Notes on Supersymmetry

Universidad de Santiago de Compostela

4 de diciembre de 2012 ´Indice general

1. Non perturbative aspects of N=1 Supersymmetric Theory 3

1.1. SU(Nc)-QCD with Nf ﬂavours...... 3 1.1.1. Field Content ...... 3 1.1.2. Action and symmetries ...... 5

1.2. SQCD Classical Moduli Space: M0 ...... 6 1.2.1. Classical Moduli Spaces ...... 6

1.2.2. M0(SQCD) ...... 6 1.3. SQCD, Quantum Eﬀective Action ...... 9 1.3.1. SQCD: Renormalization Group Analysis ...... 12 1.3.2. Holomorphic β function ...... 15 1.3.3. Matching Conditions ...... 18 1.3.4. Konishi Anomaly ...... 19 1.4. Non Perturbative Superpotentials ...... 26

1.4.1. Nf < Nc massless: The Aﬄeck-Dine-Seiberg Superpotential . . . . . 26

1.4.2. Nf < Nc mass deformation ...... 29 1.4.3. Integrating Out and In ...... 32

1.4.4. Nf ≥ Nc: The quantum moduli space ...... 36 1.5. Seiberg Duality ...... 39 1.5.1. Duality in the Conformal Window ...... 42 1.6. Conifold Field Theory (Klebanov-Witten) ...... 44 1.6.1. Free Theory ...... 44 1.6.2. Adding a Superpotential ...... 45 1.6.3. Conifold Connection ...... 46 1.6.4. Global Symmetries ...... 46

2. Selected Topics in Gauge Theories 49 2.1. Anomalies in Gauge Theories ...... 49

1 3. The Minimal Supersymmetric Standard Model 51 3.1. Introduction ...... 51 3.2. The MSSM ...... 52 3.2.1. Field content ...... 52 3.2.2. Couplings ...... 53 3.2.3. R-parity ...... 54 3.3. Electroweak symmetry breaking ...... 55 3.4. Supersymmetry Breaking ...... 57

2 Cap´ıtulo 1

Non perturbative aspects of N=1 Supersymmetric Theory

1.1. SU(Nc)-QCD with Nf ﬂavours.

Let us start by deﬁning the class of microscopic models we will be dealing with in this course. They represent the minimal supersymmetric extension of QCD, and hence receive the generic name of SQCD.

1.1.1. Field Content

Chiral Matter. i A single flavour is composed of a pair of chiral superfields (Q , Q˜i), i = 1, ..., Nc. With yµ = xµ − iθσµθ¯ √ Qi(x, θ, θ¯) = φi(y) + 2θψi(y) + θ2F i(y) √ 2 Q˜i(x, θ, θ¯) = φ˜i(y) + 2θψ˜i(y) + θ F˜i(y) The component fields receive the generic names of squark (φi), and quark fields i ˜ (ψα, ψα i). Gauge Fields a † In the Wess Zumino gauge V = TaW = V can be expanded as 1 V a(x, θ, θ¯) = θσµθA¯ a (x) + iθ2(θ¯λ¯a) − iθ¯2(θλ)a + θ2θ¯2D2(x) (1.1) µ 2 a ¯aα˙ a in terms of a gaugino (λα, λ ), and a gauge boson Aµ. The gauge invariant field strength 1 W (x, θ, θ¯) = − D¯ 2e−2V D e2V (1.2) α 4 α a a µν a 2 µ ¯α˙ a = −iλ (y) + θαD (y) − i(σ θ)αFµν(y) − θ σαα˙ (Dµλ ) (y) Ta (1.3)

3 is a chiral superﬁeld, i.e. DW¯ = 0. From here

2 1 µν a µ ¯ 2 i ∗ µν TrW 2 = − F Fµν + 2iλ σ Dµλa + D + F F (1.4) θ 2 2 µν hence

1 2 1 µν i µ ¯ 1 2 (TrW 2 + h.c.) = − F Fµν + λσ Dµλa + D 4g2 θ 4g2 g2 2g2 θ 2 θ ∗ µν −i (TrW 2 − h.c.) = F F (1.5) 32π2 θ 32π2 µν with 1 F ∗ = F αβ µν 2 µναβ Fµν = ∂µAν − ∂νAµ + i[Aµ,Aν] α˙ α˙ α˙ Dµλ¯ = ∂µλ¯ + [Aµ, λ¯ ] (1.6)

The gauge transformation of the different fields that enter is given in terms of a chiral i a i † †a † †a superfield Λ j = Λ (Ta) j and its antichiral hermitian conjugate Λ = Λ Ta = Λ Ta i - Q transforms in the Nc

i −2iΛ i j † † 2iΛ† i Q → (e ) jQ ; Qj = Qi (e ) j (1.7)

1 - Q˜i transforms in the N¯c, or

2iΛ i †i −2iΛ† i † j Q˜i → Q˜i(e ) j ; Q˜ → (e ) jQ˜ (1.8)

2V - e transforms in the adjoint (Nc, N¯c) representation

† † e2V → e−2iΛ e2V e2iΛ ; e−2V → e−2iΛe−2V e2iΛ (1.9) so that the combination Q†e2V Q + Qe˜ −2V Q˜† is gauge invariant. Also for the chiral field α˙ iΛ iΛ¯ strength, using [D¯ , e ] = [Dα, e ] = 0 we find 1 W = − D¯ 2e−2V D e2V =⇒ W → e−2iΛW e2iΛ α 8 α α α 1 † † W¯ = − D2e2V D¯ e−2V =⇒ W¯ → e−2iΛ W e2iΛ (1.10) α˙ 8 α˙ α˙ α 1Remember that given a representation D(g)D(g0) = D(gg0) of a group, automatically we have two new ones, D(g)∗ and D(g)−1t which in principle are inequivalent. The associated Lie algebra D(g) = i t −1t t i 1 + iαiD(L ) + ... has correspondingly inequivalent representations −D , D(g) = 1 − iαiD (L ) + ... and ∗ ∗ i i i t i ∗ D(g) ∼ 1 − iαiD (L ) + ... . With D(L ) hermitian D(L ) = D(L ) , these are equivalent only for real parameters αi ∈R. However, for general gauge transformations the parameter functions in (1.7) are chiral ∗ −1t superfields Λ 6= Λ . Hence by N¯c we must specify what we mean, and the choice is D , whence (1.8).

4 1.1.2. Action and symmetries

The action can be written in superspace notation as follows

τ Z 1 Z L = Im Tr d2θW 2 + d2θd2θ¯ Q† e2V Qf + Q˜ e−2V Q˜†f 8π 4 f f (1.11) Z 2 f + ( d θ W(Q , Q˜f ) + h.c.)

fi where we have extended our chiral superﬁelds to a set (Q , Q˜fi), i = 1, ..., Nc; f = 1, ..., Nf , 4πi θ τ = + (1.12) g2 2π hence the gauge-kinetic term reads as follows

Im Z 1 Z θ Z τ Tr d2θW 2 = dθ2 TrW 2 + h.c. − i dθ2TrW 2 − h.c. 8π 4g2 32π2 1 i 1 θ = Tr − F F µν + λσµD λ¯ + D2 + F˜ F µν(1.13) 4g2 µν g2 µ 2g2 32π2 µν

The superpotential contains typically mass as well as higher interaction terms

˜ g fi ˜ fi gj hk ˜fi ˜gj ˜hk W(Q, Q) = mf Q Qgi + afghijkQ Q Q +a ˜fghijkQ Q Q + ... (1.14) where afgh, a˜fgh are SU(N) invariant tensors proportional to fgh [1] p.31. However these higher terms violate baryon number conservation. 1.1.2.1 Scalar Potential

2 Nc −1 X a 2 X 2 2 V = |D | + (|F f | + |F ˜ | ) Q Qf a=1 f N 2−1 ! 1 Xc X ∂W ∂W¯ = |φ∗f T aφ − φ˜ T aφ˜∗f |2 + + (φ ↔ φ˜) (1.15) 2 f f ∂φfi † a=1 f,i ∂φfi

1.1.2.2 Global symmetries

In the absence of superpotential (W = 0) → U(Nf ) × U(Nf ) × U(1)R ∼ SU(Nf ) × SU(Nf ) × U(1)B × U(1)R × U(1)A. SU(Nf )L SU(Nf )R UB(1) UA(1) UR(1) V 0 0 0 0 0

f Q Nf 0 1 1 0

Q˜f 0 N¯f -1 1 0

5 U(1)R is a geometrical symmetry

(θ, θ¯) → (eiαθ, e−iαθ¯), (dθ, dθ¯) → (e−iαdθ, eiαdθ¯) thus although the chiral superfields are neutral, the higher components inside are charged. In the presence of a superpotential of the form (1.14) and vanishing trilinear couplings, the global symmetry reduces to SU(N)V × U(1)B in the case of diagonal equal masses fg f f m = mδ g and to U(1)B for generic m g. 1.1.2.3 Theta term The gauge kinetic term (1.13) contains a θ-term. Remember that θ is a periodic variable. In other words θ → θ+2π is a symmetry of the quantum theory. This is because spacetime integral of this term computes the“winding number” of the gauge field at infinity. Namely, there exist non-trivial gauge field configurations for which θ Z d4xFF˜ = nθ (1.16) 32π2 with n an integer. Thus, although strictly speaking θ → θ + 2π is not a symmetry of S, it shifts trivialy the phase factor in the path integral.

1.2. SQCD Classical Moduli Space: M0

1.2.1. Classical Moduli Spaces

A common feature of supersymmetric models is the existence of a large amount of vacua, spanning continuous manifolds which generically receive the name of “moduli space”. Moreover, as we shall see, quantum corrections do not generically lift this degeneracy of vacuum states. This is unlike non-supersymmetric theories, for which vacua are usually discrete sets of points. The vacuum condition is simply V = 0. The space M0 is in many cases not a manifold, but rather a set of manifold or ”branches”, often joined along subspaces of lower dimension. As an example consider a model with 3 chiral multiplets X, Y and Z, and a superpotential W = λXY Z. The susy vacuum condition states that xy = xz = yz = 0 for the scalar component. So we ﬁnd 3 branches, those are (y = z = 0 with x arbitrary) and cyclic permutations x → y → z → x of the same statement. The three branches join at a single point x = y = z which is a singular point of W (all its gradients vanish).

1.2.2. M0(SQCD)

Let us start by examining this manifold in the case of the lagrangian (1.11) with a vanishing superpotential to start with. Later on we will consider the quantum modiﬁcation of this lagrangian, and we will show that nonperturbative eﬀects allow for the appearance of a nontrivial W.

6 With W = 0, the vacuum conditions can be read from setting V = 0 in (1.15), and therefore they also are called D-ﬂatness conditions

X a 2 X ∗f a ˜ a ˜∗f 2 |D | = |φ T φf − φf T φ | = 0 (1.17) a a

The solution to these algebraic equations fall into three categories, whether Nf < Nc, Nf = Nc, Nf = Nc + 1 or Nf > Nc + 1. Let us examine them case by case.

fi ˜ Nf < Nc: Performing an SU(Nf ) × SU(Nc) rotation we may bring φ and φfi to the form Nc z }| { a1 0 ··· 0 ···  fi fi  0 a2 ··· 0 ···  φ = af δ =   Nf (1.18)  ···············   0 0 ··· aNf ··· ˜ and the same for φfi =a ˜f δfi. Hence (1.17) yields

This has the generic solution

|af | = |a˜f |, f = 1, ..., Nf . (1.20)

For generic af 6= 0, f = 1, ...Nf SU(Nc) breaks down to SU(Nc − Nf ). Now the question is what kind of quantity describes this D-ﬂat moduli space. In other words, what degrees of freedom are the relevand ones at low energies. Since a spontaneo- 2 us symmetry breaking is at work the natural guess is that of Nf gauge invariant composite chiral “meson”superﬁelds (Goldstone supermodes)

f fi M g = Q Q˜gi, f, g = 1, ...., Nf . (1.21)

whose vacuum expectation values parametrize the vacuum manifold.

hM f i = diag(a2, a2, ..., a2 ). (1.22) g 1 2 Nf

f The counting matches. We started with 2NcNf massless chiral superﬁelds Q and ˜ 2 f Qg, and ended up with Nf modes M g. The diﬀerence is the number of degrees of freedom (Goldstone modes) that have been eaten up in the Higgs mechanism, which is also the number of particles becoming massive

SU(Nc) SU(Nc−Nf ) Higgsed modes z 2}| { z }| 2 { z }| 2{ (Nc − 1) − (Nc − Nf ) − 1 = (2NcNf − Nf ) (1.23)

7 Nf ≥ Nc. In this case, the best one can do is bring the chiral superﬁelds to the form

Nc z }| { a1 0 ··· 0   0 a2 ··· 0    fi fi  ············  φ = aiδ =   Nf (1.24)  0 0 ··· aN   f   0 0 ··· 0   ············  and the same for φ˜gi =a ˜gδgi. In this case the D-ﬂatness condition (1.17) is the same PNc a i but now with Nf ≥ Nc. Given the tracelesness of the gauge generators i=1(T ) i = 0, the generic solution of (1.203) is given by

2 2 2 |af | − |a˜f | = v , ∀f = 1, ..., Nf (1.25)

Now the gauge group is completely broken at a generic point on moduli space. The gauge invariant condensates that can be formed are now

f fi M g = Q Q˜gi (1.26)

B[f1...fNc ] = Qf1i1 ... QfNc iNc (1.27) i1...iNc

B˜ = Q˜ ... Q˜ i1...iNc . (1.28) [f1...fNc ] f1i1 fNc iNc

The“baryon”indices are automatically antisymmetrized in ﬂavor space. In total they are 2 Nc 2 2Nf ! Nf + 2CN = Nf + (1.29) f Nc!(Nf − Nc)! quantities, which exceeds the number of initial chiral modes minus the number of 2 broken generators 2NcNf − (Nc − 1). The clue is that the previous quantities are in some cases redundant, as can be seen by ﬁnding agebraic identities that link them. Let us see this in the special cases Nc = Nf and Nc = Nf + 1. fi ˜ 2 Nf = Nc The matrices Q and Qfi are square matrices, and hence Nf + 2 gauge invariant polynomial can be formed

f f fi M g = Q Q˜g ,B = det Q , B˜ = det Q˜fi . (1.30) However they are linked by the following trivial identity

det M = BB˜ (1.31)

2 So we have really Nf + 1 moduli. Let us check the counting: SU(Nc) is completely 2 2 broken, hence Nc − 1 = Nf − 1 Goldstone modes have been eaten up. Since we 2 2 started from 2NcNf = 2Nf , the diﬀerence yields precisely Nf + 1 moduli, which are the ones in (1.30) modulo the constraint (1.31).

8 2 Nf = Nc + 1 In this case there are Nf + 2Nf gauge invariant quantities It is convenient to exhibit the bayonic operators (1.27) and (1.28) in the Hodge dual form

1 B¯ = Qf1i1 Qf2i2 ··· QfNc iNc (1.32) g gf1···fNc i1i2···iNc Nc!

¯ g 1 B˜ = gf1···fNc Q˜ Q˜ ··· Q˜ i1i2···iNc (1.33) f1i1 f2i2 fNc iNc Nc!

f f M g = Q˜ Qg (1.34)

2 Nc→Nf −1 2 which exceeds by 2Nf the number of moduli 2NcNf − (Nc − 1) = Nf . Again they are not independent but satify the equalities

f f ¯ g −1 g ¯ g B¯f M g = M gB˜ = 0 ; det M(M )f = B¯f B˜ (1.35)

Indeed 1 B¯ M f = Qf1i1 Qf2i2 ··· QfNc iNc Qfj Q˜ = 0 (1.36) f g ff1···fNc i1i2···iNc gj Nc!

and this vanishes because the set of indices [i1, ..., iNc , i] is antisymmetric and necessarily two of them are repeated. In the second equation, the left hand side stands g short for the minor of the element M f i.e. the determinant of M with the f’th row and g’th column deleted. This is precisely 1 l.h.s. = g1···gNc f M f1 ··· M fNc f1···fNc g g1 gNc Nc! 1 = Qf1i1 ··· QfNc iNc g1···gNc f Q ··· Q f1···fNc g g1i1 gNc iNc Nc!

= Qf11 ··· QfNc Nc g1···gNc f Q ··· Q f1···fNc g g11 gNc Nc

¯ f = B¯gB˜ = r.h.s (1.37)

1.3. SQCD, Quantum Eﬀective Action

The vacuum is the lowest energy state of the theory. The proper tool to investigate it when quantum fluctuations are taken into account is the quantum effective action. There are two concepts for such object. The standard textbook in QFT introduces it as the gene- rating functional of 1PI correlation functions. In a more modern approach the Wilsonian effective action links with the theory of critical phenomena and renormalisation group analysis. Whereas the first one is plagued with infinities and must be subject to a proper

9 renormalisation program, the second is finite and well defined by construction, hence we will stick to it in what follows. Defining a quantum field theory starts by giving a classical local action which describes the degrees of freedom and the classical dynamics. Central to the discussion are two concepts: the coupling constants and the cutoff scale. Without a cutoff scale, a QFT is ill defined and plagued with infinities. The classical renormalisation program is a way to handle these divergences. Another approach is that of a Wilsonian effective action. A Wilsonian effective action has the following generic aspect Z d X S(µ) = d x gi(µ)Oi (1.38) i were Oi label an infinite set of local operators (polynomials in basic fields and derivatives thereof) A field theory is normally defined by specifying the bare parameters in a given action gi(µ) at some cutoff scale µ. One then makes use of this action to seed a path integral that will produce Green’s functions which will be calculable in terms of gi(µ). Imagine |pµ| ∼ E ≤ µ is the typical (euclidean) momentum scale of the incoming particles in the process. Then we loosely refer to these amplitudes as Γ(E, gi(µ)). Fixing these amplitudes to experimental values is a practical means of fixing the values of the couplings gi(µ). The actual computation of Γ(E, gi(µ)) involves integrating over loop momenta pµ in the range E ≤ |pµ| ≤ µ. This typically introduces logarithms log(µ/E). Hence on one side, for E = µ the tree level result is exact, and on the other hand for E << µ the logarithms grow larger than the tree level values and spoil the perturbative analysis. It may therefore be convenient to define the theory at a different scale µ0 < µ such that the tree level results better approximate the physics. This is the task of the renormalization group, which answers to the following question: how must the parameters change in order for the physics at scale E to remain intact? 0 0 0 Γ(E, gi(µ), µ) = Γ(E, gi(µ ), µ ) (1.39)

The dependence of gi on the deﬁning scale µ is encoded in the Renormalization Group Equation (RGE) dg (µ) µ i = β (g (µ), µ)) (1.40) dµ i i As an example consider the perturbed free scalar ﬁeld theory Z 1 S[φ] = ddx ∂φ∂φ (1.41) free 2 Clearly for this action to be invariant under µ → µ0 = λµ and x → x0 = λ−1x where λ < 0 (hence x growing in length) φ must scale as φ → λφ. Hence the mass dimension [φ] = +1. d Consider an operator made out of φ and derivatives thereof such that Oi → λ i Oi. We say that Oi has classical (mass) dimension di. Next we write the perturbed action as follows " # Z Z(µ) X S[φ; µ, g ] = ddx ∂ φ∂µφ + g (µ)O (x) (1.42) i 2 µ i i i

10 For the action to be dimensionless, hence invariant under the previous scaling, the coupling d−d constants must have classical dimension d − di. We write gi(µ) = µ i g˜(µ) whereg ˜(µ) is dimensionless. Hence we have

dgi dg˜i(µ) µ = (d − d)µdi−dg˜ (µ) + µd−di µ dµ i i dµ quant = (di − d)gi(µ) + βi = βi (1.43)

Of particular importance are the limits µ → ∞ (UV) and µ → 0 (IR). If there exist a sensible limit where the couplings don’t run away, then the theory must reach a fixed ∗ point g(µ → 0) = gi . Theories at a fixed point are very special because they are scale and even conformal invariant. They have neither dimension-full parameters nor massive states. They are called conformal field theories (CFT). ∗ In the neighbourhood of a fixed point CFT we have gi = gi + δgi and we can always linearise the RG flows dgi 2 µ = Aij δgj + O(δgj ) (1.44) dµ ∗ gj +δgj and go to a diagonal basis δgi → δg˜i, Aij → ∆iδij such that , to linear order dδg˜ µ i = (∆ − d)δg˜ = βquant(g∗) (1.45) dµ i i i and so, to linear order the RG flow is simply

µ ∆i−d δg˜ (µ) = δg˜ (µ0) (1.46) i µ0 i

The quantity ∆i is called the scaling (or conformal) dimension of the operator associated to δg˜i. It will be different, in an interacting QFT, from the classical scaling dimension di, actually ∆i = di + γi (1.47) where is called the anomalous dimension of the operator and its origin is purely quantum. From (1.45) we see that, close to the fixed point, the couplings can be classified in the following way

Relevant. If ∆i < d the coupling decreases (increases) towards the UV (IR). So in the vicinity of a UV ﬁxed point (µ → ∞) all these couplings vanish.

Irrelevant. If ∆i > d we have just the opposite of the above. Hence as we approach a IR ﬁxed point (µ → 0) all irrelevant couplings disappear, and only relevant couplings remain.

Marginal. The case ∆i = d signals couplings which stay fixed under the RG flow. However for them, the linearised analysis is not valid and one has to go beyond leading order. If finally the coupling does not run to all orders, we speak of a truly marginal coupling.

11 In summary, when building the low energy effective action, which capture the IR physics, only relevant operators with βi < 0 are important. Moreover, they become negligible in the UV, so the actual value they have there is actually of very little importance. The number of relevant operators is finite, and actually very small in a supersymmetric theory. From the perturbative point of view, relevant operators are the same as renormalizable ones. So those for which the coupling constants have scaling dimension ∆i < 4. In a first approximation, we can trade the full scaling dimension for the classical one ∆i ∼ di. If we were talking about a scalar field theory this would leave us with an action of the form

Z 1 S = d4x (∂φ)2 + m2φ2 + βφ3 + λφ4 (1.48) 2

In superspace the counting is a little bit diﬀerent. This is because the anti commuting coordinates have scaling dimension [θ] = 1/2 but [dθ] = −1/2, given that R dθθ = 1. Hence, for an action like (1.11), appart from the kinetic terms, we are speaking of superpotential terms which have at most dimension 3

i j i j k Wrel = mijΦ Φ + λijkΦ Φ Φ . (1.49)

Quantum Vacua

In QFT, scalar fields may develop a non-trivial vacuum expectation value (VEV), hφii = 0. The set of all VEV’s characterises the vacuum state itself, and are determined by the requirement that they are minima of the low energy effective potential. The low energy effective potential is the potential of the WIlsonian effective action in the limit µ → 0, hence after all non-zero modes of the fields have been integrated out.

1.3.1. SQCD: Renormalization Group Analysis

From (1.11) we see that typically the supersymmetric version of QCD consists of three types of terms. Let us repeat here the tree level action, stressing the fact that it is the UV action deﬁned at scale µ0

τ(µ ) Z Z L(µ ) = Im 0 Tr d2θW 2 + Z(µ ) d2θd2θ¯(Q†e2V Q + Qe˜ −2V Q˜†) 0 8π 0 (1.50) Z 2 + ( d θ W(Q, Q˜; λi(µ0)) + h.c.)

We have displayed the parameters that will depend on the UV renormalization scale, which we have called µ0. Actually it is conventional to set Z(µ0) = 1 by deﬁnition as an initial condition.

Perturbatively, the fate of the diﬀerent terms along RG ﬂow µ0 → µ is condensed in the following collection of statements

1. The couplings λi(µ) are invariant, hence receive no corrections.

12 2. The function τ(µ) receives corrections at 1-loop in perturbation theory.

3. The function Z(µ) receives corrections from all loops in perturbation theory.

1.3.1.1 Non-renormalization theorem From these, the most famous one is the last one, which usually goes under the name of “non-renormalization theorem”, stating that the superpotential is not renormalized in perturbation theory. Let us give here a proof based on symmetry arguments [2]. The prototypical example deals with the Wess-Zumino lagrangian Z Z S = d4θ ΦΦ† + d2 θ(mΦ2 + λΦ3) . (1.51)

There is no U(1)R charge assignement that makes this lagrangian invariant. However one can play the trick of envisaging the constants m, λ as background (spurion) ﬁelds that have acquired a v.e.v. The following charges do the job

U(1)R U(1)B

Φ 1 1 (1.52) m 0 -2 λ -1 -3

Under RG ﬂow, the K¨ahler form and the superpotential will run over to some unknown functions

K = K(Φ, Φ†, m, m∗, λ, λ∗) W = W(Φ, m, λ) (1.53)

In a Wilsonian setup, integrations over ﬁnite momentum intervals cannot spoil holomorphicity of W. This, together with the requirement that the eﬀective lagrangian has the same global symmetries restricts the form of W as

λΦ X W = mΦ2f = a λnm1−nΦn+2 (1.54) m n n The existence of a weak coupling limit λ → 0 restricts n ≥ 0, and continuity of the massless limit m → 0 implies n ≤ 1, thus

2 3 W(µ) = mΦ + λΦ = W(µ0) (1.55)

1.3.1.2 Perturbative Non-Renormalization theorem Perturbation theory can be implemented in full superspace, by means of the supergraph computation. Each such supergraph computes a contribution to the 1PI Eﬀective Action. An important point to stress, is the fact that all such contributions come out with an integral over full superspace R d2θd2θd¯ 4p..... In other words, superspace perturbation theory only produces D terms, and never F terms. Hence it will

13 be only capable of giving contributions that correct the K¨ahler potential, but never the superpotential. So simple is in essence the content of the famous nonrenormali- zation theorem. Of course this only means that the parameters in the superpotential τ, m2, λ, ... do not have renormalizations Z’s, other that the ones than come from the

wave function renormalization ZQ and ZQ˜ . 1.3.1.3 Holomorphic Anomalies In supersymmetric gauge theories and, in general, in theories with massless particles that can propagate inside the loops, some propagators may lead to infrarred singularities in the limit kµ → 0. This oﬀers a mechanism to evade the perturbative non-renormalization theorem. Consider the following D-looking-term

Z D2 I = d4xd2θd2θ¯ F (Φ) (1.56) 2 ¯ 2 By using the identity [D , D ] ∼ we may perform the following manipulations, (neglect total derivatives)

Z 2 4 1 α ¯ 2 1 D I = d x( D D Dα − ) F (Φ) 16 4 θ=θ¯=0 Z ¯ 2 2 1 4 α D D = d xD DαF (Φ) 16 θ=θ¯=0 Z ¯ 2 2 1 4 α [D ,D ] = d xD DαF (Φ) 16 θ=θ¯=0 Z 1 4 2 = d xD F (Φ) 16 θ=θ¯=0 1 Z = − d4xd2θF (Φ) (1.57) 4 The end result goes into a correction of the superpotential. However this situation is to be expected only in 1PI effective actions. In Wilsonian effective actions the integrals in momentum are cutoff, above by the UV definition scale µ0 and below by the renormalization scale µ. In conclusion, they never will generate a singular contribution like (1.56). Thus, we state that a Wilsonian superpotential is never generated in per- tubation theory. At the non-perturbative level, i.e. functions that admit an expansion 2 in powers of e1/g , the possible answers are constrained by the global symmetries. Flavour symmetries SU(Nf ) × SU(Nf ) impose to use as variables singlets like det M, B and B˜. U(1) symmetries are more delicate, since care of chiral anomalies must be taken.

1.3.1.4 Anomalous dimensions from wave function renormalization

It is important to stress that the non-renormalization theorem for the couplings λi in the superpotential has been derived under the assumption that the kinetic term is not canonically normalized. Contrarily to the superpotential, the K¨ahler potential, not being protected by holomorphy can, and indeed will, receive contributions from higher loops that will correct it multiplicatively

∗ ∗ K(µ) = Z(m, m , λ, λ , µ)K(µ0). (1.58)

14 In order to compare theories along the RG trajectory it is always convenient to stick to normalized kinetic terms for all the fields. In particular, for the chiral superfields this involves a redefinition

1 ˜ 1 ˜ Q = p Qc ; Q = p Qc . (1.59) Z(µ, µ0) Z(µ, µ0)

In terms of these (Qc, Q˜c) the kinetic terms will always stay canonically normalized τ(µ) Z Z L(µ) = Im Tr d2θW 2 + d2θd2θ¯ Q†e2V Q + Q˜ e−2V Q˜† 8π c c c c Z 2 ˜ c + ( d θ W(Q, Q; λi (µ)) + h.c.) (1.60) (1.61)

However, now the new couplings are no more RG invariant. For example for the mass term c mQQ˜ = m QcQ˜c we would have

c −1 m (µ) = m Z(µ, µ0) (1.62) and from here the anomalous dimension totally arises from the wave function renormalization c dm (µ) c d log Z(µ, µ0) c β c ≡ = −m (µ) ≡ −m (µ)γ(µ) (1.63) m d log µ d log µ 1.3.1.5 Perturbative Quantum Moduli Space In this way we arrrive at the following important result: if we start from a bare lagrangian which has a vanishing superpotential W = 0, a Wilsonian superpotential will never be generated along a RG trajectory in perturbation theory. Stated in a different way, the classical moduli space and the quantum moduli space, given by solving the D flatness condition (1.203), are the same. Reverting the logic of the argument, if a superpotential is seen to arise along an RG- trajectory, it must necessarily come from non-perturbative effects, such as instanton configurations, which introduce corrections in the effective action weighted by e1/g2 . The possible superpotentials thus generated dynamically are only constrained by the global unbroken symmetries. Flavor symmetries SU(Nf )L × SU(Nf )R dictate that the allowed superpotentials must be functions of flavor singlet polynomials det(M), B or B˜. U(1) symmetries are more delicate as they involve chiral fermions and care must be taken of anomalies. The main aim of these lectures is the study of non-perturbatively generated superpotentials.

1.3.2. Holomorphic β function

A Wilsonian UV action at scale µ incorporates the information about all the frequencies at scales above µ. Hence it is only capable of describing physical quantities at lower energy scales E ≤ µ. The computation will tipically involve integrals ranging from µ to 0, but

15 the slice between E and 0 can be included in a renormalization scheme. In this scheme, the tree level form of the UV Wilsonian action at scale µ describes the physical processes at energy E ' µ, up to corrections of O(log µ/E). The change of the parameters when µ → µ0 by construction keeps the physics at E ≤ µ0 ≤ µ invariant.

If we take the action (1.50) as the deﬁning UV action at scale µ0, at a lower scale µ < µ0, the form of the action is the same, with renormalized τ0 = τ(µ0) → τ(µ). τ(µ) Z Z L(µ) = Im Tr d2θW 2 + Z(µ) d2θd2θ¯(Q†e2V Q + Qe˜ −2V Q˜†) 8π Z 2 + ( d θ W(Q, Q˜; λi(µ0)) + h.c.) (1.64) where we have taken account of the non-renormalization theorem for the superpotential automatically. Under and RG ﬂow, µ0 → µ < µ0 the gauge coupling constant gets renormalized as τ(µ) = τ(µ0) + f(τ0; µ, µ0) . (1.65)

In order to get hold on the possible form for f(τ0; µ, µ0) we take into consideration the following constraints

holomorphicity: by which we mean that f(τ0; µ, µ0) is a holomorphic function of τ0.

periodicity: under shifts θ → θ + 2πi ⇒ τ0 → τ0 + 1. Then at most f(τ0 + 1; µ, µ0) = f(τ0; µ, µ0)+n(µ, µ0) where n(µ, µ0) ∈ Z. Given the boundary condition n(µ0, µ0) = 0 by continuity we arrive at n(µ, µ0) = 0. This means f(τ0; µ, µ0) is a periodic function under shifts θ → θ + 2πi, or f(τ0 + 1; µ, µ0) = f(τ0; µ, µ0) and we may expand in Fourier series ∞ X µ τ(τ ; µ, µ ) = τ + f e2πniτ0 (1.66) 0 0 0 n µ n=0 0

0 0 transitivity: f0(µ , µ0) = f0(µ , µ) +f0(µ, µ0). This forbids to a logarithm f0(µ, µ0) = log(µ/µ0).

In summary, fulﬁlling all the constraints yields an almost unique answer, and we get the important result that the perturbative contribution is saturated at one loop µ ib µ τpert( , τ0) = τ0 + ln . (1.67) µ0 2π µ0 4πi θ and with τ = g2 + 2π , we obtain the perturbative running of the coupling constant

1 1 b µ 2 = 2 + 2 log (1.68) g (µ) g (µ0) 8π µ0 where b is the coeﬃcient of the 1-loop beta function in the usual RG equation dg(µ) b µ = − g3 (1.69) dµ 16π2

16 which must be computed explicitely. One may infere it from the standard QCD result and the ﬁeld content in (1.50).

11 1 X 1 X b = T (A) − T (R ) − T (R ) (1.70) 6 3 fer 6 sc fer sc where “fer” runs over Weyl fermions and “fer” over complex scalars. In N = 1 SYM, f ˜ “fer” will count the gauginos λ in the adjoint (R = A) and the 2Nf Weyl quarks ψ , ψf ,, f ˜ whereas “sc”will run over the 2Nf squarks φ , φf . Therefore we ﬁnd

2Nf 11 1 X 1 1 b = − T (A) − + T (R ) 6 3 3 6 f f=1

Nf 3 X = T (A) − T (R ) (1.71) 2 f f=1 where T (R) = C(R)/C(F ) is the index of the representation R, and C(R)δab = tr(R(ta)R(tb)). Tipically C(F ) = 1/2 and C(A) = N for SU(N), hence

1 R = F = N or N¯ of SU(N ) T (R) = c c c (1.72) 2Nc R = A = adjoint of SU(Nc) and so we obtain for b

Nf X b = 3C(A) − C(Rf ) = 3Nc − Nf . (1.73) f if all the matter falls in the fundamental representation. From (1.68) we derive two known results.

Diﬀerentiating with rexpect to log µ we obtain the holomorphic perturbative beta function for SU(Nc) with Nf ﬂavours

g3 β(g) = − (3N − N ) (1.74) 16π2 c f

Deﬁning µ = Λ as the scale where g(Λ) = ∞ gives

2 b b 8π Λ = µ0 exp − 2 . (1.75) g (µ0)

Making use of (1.69) with µ = µ0 one can easily check that dΛ/dµ0 = 0. Hence, Λ does not depend on µ0 itself, but rather on the value of the bare coupling g(µ0) at the UV scale µ0. This trading of a dimensionless quantity, g(µ0), by a dimensionfull

17 number, Λ, goes under the name of dimensional transmutation. It will turn out to be consistent to extend the deﬁnition of Λ to complex values,

Λ b 8π2 2πiτ(µ0) = exp − 2 + iθ = e (1.76) µ0 g (µ0)

1.3.2.1 Absence of non-perturbative corrections for pure N = 1 SYM We have seen in (1.67) that the perturbative running is exhausted by the one-loop contribution. In the case of pure N = 1 SYM, A slight reﬁnement of this argument permits to show that potential non-perturbative corrections shown in (1.66) are also absent. In N = 1 SYM there are massless Weyl fermions, and as we will soon see, anomalous global U(1) symetries. In such a context, the θ parameter is as unphysical as the choice of the origin of phases for the Weyl fermions. Therefore there should be a consistent reduction of the RG ﬂow to θ = 0. However it is evident from (1.66) that setting Re(τ0) = 0 we do not get Reτ(µ) = 0 except if fn = 0 for n ≥ 1. In contrast, the n = 0 piece (1.67) leaves θ intact.

1.3.3. Matching Conditions

The previous analysis allows us to explore the behaviour of the eﬀective action when the scale is moved relative to the value of the relevant scales in the theory. The later are set by dimensionfull couplings, like masses, and by vevs. Consider the following two possible scenarios

Suppose we start from an SU(Nc) theory with Nf ﬂavors. From (1.203) we learn

that we can give to one of the chiral fields, a vev, say |aNf | = |aÑf | = v. With large momentum scales q >> v the theory looks like SU(Nc) with Nf flavors, but for q << v the theory looks like SU(Nc − 1) with Nf − 1 flavors. Each regime has its own dynamical scale Λ. This setup can be generalized to an arbitrary number of vevs. |ag| = |a˜g| = v for g = 1, ..., ∆). Alternatively, we may imagine that a mass term for a given quark is introduced

Lm = mQQ˜

For energies p >> m the theory looks like (SU(Nc),Nf ). However much below p << m the quanta of this ﬁeld are not excited and we may replace it by its classical vev, which actually is vanishing, as imposed by the F term conditions: ˜ FQ = mQ = 0 ; FQ˜ = mQ = 0

Therefore the ﬁeld completely disappears, and the theory looks eﬀectively like (SU(Nc),Nf − 1).

To cope with both possibilities consider the general problem of matching of high and low energy theories s (SU(Nc),Nf ) ,→ (SU(N˜c), N˜f ) (1.77)

18 with N˜c = Nc − ∆ and N˜f = Nf − Ξ, where s stands for a mass scale, either v or m, where both, high and low energy-theories must connect. Each theory has its own RG equation dynamical scale

2πiτ(µ) Λ(µ) = µe b 2πiτ˜(˜µ) Λ(˜˜ µ) =µe ˜ ˜b (1.78)

In principle, the deﬁnition scales µ andµ ˜ are independent. However in this case they are linked by the fact that the eﬀective theory and the microscopic theory match together at scale s ˜ ˜ τ(s) =τ ˜(s) ⇒ Λb = Λ˜ bsb−b (1.79)

1.3.4. Konishi Anomaly

The Konishi anomaly is the supersymmetric version of the familiar axial anomaly. Consider to the following super-chiral transformations Q → Q0 with

0f iα f ˜0 iβ ˜ Q = e Q ; Qf = e Qf 0†f −iα ¯f ˜0† −iβ ˜† Q = e Q ; Qf = e Qf (1.80) with α, β real parameters. This is by itself a (global) symmetry of the ungauged action Z S(Q, Q†, Q,˜ Q˜†; V ) = d4xd4θ (Q†e2V Q + Qe˜ −2V Q˜†) Z Z + d4xd2θW(Q, Q˜) + d4xd2θW¯ (Q†, Q˜†) (1.81) for vanishing superpotential W = 0. Considering the action of this symmery on the fer- ˜ f f mions ψ and ψ inside Q and Q˜ for α = β this is an extension of the familiar U(1)A axial symmetry, whereas for α = −β this is the analog of the vector (or baryon) U(1)V symmetry. Now in order to capture the associated Ward identity we promote this global transformations to local ones. Moreover we will relax the reality conditions on α and β and promote them to independent chiral superfields α(x, θ, θ¯) and β(x, θ, θ¯). Then the classical action will no more be invariant (unless we not transform the gauge field to compensate for it, in which case we are actually gauging the symmetry). An additional ingredient is that we shall treat all chiral fields Q, Q†, Q˜ and Q˜† as independent from one another (notice that, for example, the scalar φ in Q is now complex, hence we have doubled the degrees of freedom). Associated with each independent transformation of each field, we shall find an anomalous Ward identity. For example, increasing just ¯ Q → eiα(x,θ,θ)Q (1.82)

19 leads to 2 δS(eiαQ) δ S = iα(x, θ, θ¯) α δ iα(x, θ, θ¯) ! 1 ∂W(Q, Q˜) = iα(x, θ, θ¯) − D¯ 2Q† e2V Qf + Qf (1.84) 4 f ∂Qf

Making α ∈ R global does not make the action invariant, so we may not derive a classical conservation equation 3. However the above transformation (1.82) is by itself a quantum symmetry of the path integral Z(Q, Q†, Q,˜ Q˜†) since the quantum ﬁeld Q is integrated over and (1.82) is just a change of variables. The measure of integration changes by a jacobian, (DQf ) → (DeiαQf ) = (DQf )J(α) (1.85) whose covariant-superspace evaluation was performed by Konishi and Shizuya [4], with the result Z T (R ) J(iα) = exp − d4xd2θ f (iα) TrW 2 (1.86) 32π2 Demanding that Z be independent of α, Z(eiαQ) = Z(Q) leads now to the Q-Ward identity * + 1 ∂W(Q, Q˜) X T (Rf ) 0 = − D¯ 2 Q† e2V Qf + Qf − TrW 2 (1.87) 4 f ∂Qf 32π2 f

† −iα†(x,θ,θ¯) † the Q → e Qf gives the Jacobian

Z T (R ) J¯(−iα†) = exp − d4xd2θ f (−iα†) TrW¯ 2 (1.88) 32π2 and from here the Q† Ward identity

* † † + 1 ∂W¯ (Q , Q˜ ) X T (Rf ) 0 = − D2 Q† e2V Qf + Q† − TrW¯ 2 (1.89) 4 f f † 32π2 ∂Qf f

†f And similarly with Q˜f and Q˜ * + 1 ∂W(Q, Q˜) X T (Rf ) 0 = − D¯ 2 Q˜ e−2V Q˜†f + Q˜ − TrW 2 (1.90) 4 f f ˜ 32π2 ∂Qf f * † † + 1 ∂W¯ (Q , Q˜ ) X T (Rf ) 0 = − D2 Q˜ e−2V Q˜f + Q˜† − TrW¯ 2 (1.91) 4 f f ˜† 32π2 ∂Qf f

2The D¯ 2 appears from the chiral functional diﬀerentiation rule δ 1 α(y, θ0, θ¯0) = − D¯ 2δ(x − y)δ(θ − θ0)δ(θ¯ − θ¯0) (1.83) δα(x, θ, θ¯) 4

3this would need a compensating transformation, i.e. α =α ¯ = ±β = ±β¯ for a full global vector/axial transformation

20 1.3.4.1 Konishi Anomaly and Gaugino Condensate

f g Set W(Q, Q˜) = m gQ Q˜f in (1.87) and T (Rf ) = T (F ) = 1 for the fundamental representation. Then it can be written in components, the lowest one being: 1 † 2V f f ˜ g hQf e Q i = m ghφf φ i − hλλi (1.92) θ¯2 32π2 It turns out that the left hand side con be expressed as a total SUSY tranformation 1 † 2V f ¯ ¯α˙ f hQf e Q i = h √ {Qα˙ , ψ φf (x)}i (1.93) θ¯2 2 2

In the absence of spontaneous symmery breaking Q|0i = Q¯|0i = 0 leading to the following relationship among composite operator vev’s

2 f ˜ g hλλi = 32π m ghφf φ i (1.94)

This equations states in short that, as soon as some of the scalar ﬁelds acquire a vev in a massive theory they will trigger a gluino condensation. Moreover (1.94) will be exact to all orders in perturbation theory.

1.3.4.2 U(1)A axial anomaly f g 2 2 Again setting W(Q, Q¯) = Q˜gm gQ , multiplying (1.87) by D and (1.91) by D¯ and subtracting one gets4 µ 5 0 = h∂ Γµ − iM − ai (1.96) with

5 ¯ ¯ † 2V Γµ = (DσµD − Dσ¯µD) Q e Q /4 2 ˜ f g ¯ 2 † f ˜†f M = −(D (Qf m gQ ) − D (Qf m gQ iN a = − f (D2TrW 2 − D¯ 2TrW¯ 2) (1.97) 32π2 Equation (1.96) contains the full set of anomalies forming a supermultiplet of anomalies. For example the lowest component reduces to the standard axial U(1)A anomaly N ∂ (ψ¯ σ¯µψf ) = ψ˜ mf ψg + f Tr F F˜µν (1.98) µ f f g 32π2 µν Another way to see this is to perform simultaneous rotation of Q and Q† with real constant superﬁelds α = α† = α∗ The measure now changes by

iα f −iα † f † ¯ (De Q )(De Qf ) = (DQ )(DQf )J(iα)J(−iα) (1.99)

4one must make use of 2 2 µ † µ [D , D¯ ] = −i∂µ(Dσ D¯ − D¯ σ¯ D) (1.95)

21 where the total Jacobian now is T (R ) Z J(iα)J¯(−iα) = exp −iα f d4x(d2θ TrW 2 − d2θ¯ TrW¯ 2) 32π2 Im αT (R ) Z = exp f d4xd2θ TrW 2 (1.100) 8π 2π

4πi θ Bearing in mind the form of the gauge kinetic term in the action (1.50) with τ0 = 2 + , g0 2π this can be interpreted as a shift in the theta angle

θ → θ − αT (Rf ) (1.101)

Notice that this result can be blamed entirely to standard axial anomaly of the Weyl fermion ψfα in Qf . For a simultaneous rotation of Q and Q˜ the shift is by 2α.

Non-holomorphic schemes for β(g).

1. Canonically Normalized Chiral Superﬁelds

We pause here to stress that the running given in (1.68) has been obtained for a parame- trization of the Wilsonian RG as given in (1.64). Let us write the eﬀective lagrangian at scale µ (and for θ = 0

1 1 Z L(µ) = Tr d2θW 2 + h.c. 4 g2(µ) Z 2 2 † 2V −2V † +Z(µ, µ0) d θd θ¯(Q e Q + Qe˜ Q˜ ) (1.102)

1 where g2 (µ) = Re τ(µ), sometimes called, holomorphic coupling constant, runs exactly at one loop 1 1 b µ 2 = 2 + 2 log g (µ) g (µ0) 8π µ0 1 b g3 β( ) = ⇒ β(g) = − b (1.103) g2 8π2 16π2

It is important to stress that the one-loop saturation of the RG flow for 1/g2 is valid only if the Kähler term undergoes the wave function renormalization Z(µ0) → Z(µ, µ0). The Kähler potential is not a holomorphic quantity, and therefore Z is not protected. Generically it will receive contributions at all orders in perturbation theory. We may however decide to keep, at each stage in the RG trajectory, normalized kinetic terms for the chiral superfields. This amounts to a field redefinition p p Qc = Z(µ, µ0) Q, Q˜c = Z(µ, µ0) Q.˜ (1.104)

22 † 1 ¯ This is of the form (1.80) with iα = −iα = − 2 log Z(µ, µ0). Hence α(x, θ, θ) is a purely imaginary chiral superﬁeld. The quantum measure changes accordingly by

† iα −iα† † † ¯ † D(Q)D(Q ) = D(e Qc)D(e Qc) = D(Qc)D(Qc)J(iα)J(−iα ) T (R ) Z Z Z = D(Q )D(Q†) exp − f d4x iα d2θTrW 2 − iα† d2θ¯TrW¯ 2 c c 32π2 1 T (R ) Z = D(Q )D(Q† ) exp f log Z(µ, µ ) Tr d4xd2θ W 2 + h.c. c c f 4 16π2 0

Doing the same for Q˜ and Q˜† and adding up contributions, comparing with (1.102) we ﬁnd that, at scale µ, we can have a canonically normalized lagrangian Z Z 1 1 2 2 1 2 2 ¯ † 2V ˜ −2V ˜† Lc(µ) = 2 Tr d θW + h.c. + d θd θ(Qce Qc + Qce Qc) (1.105) 4 gc (µ) 4 with a non-holomorphic guage coupling

1 1 b µ X T (Rf ) 2 = 2 + 2 log + 2 2 log Z(µ, µ0) (1.106) g (µ) g (µ0) 8π µ0 16π f

From here and, using (1.103) and (1.71) we arrive at the following non-holomorphic (all loop) beta function   1 X 1 3 1 X β( ) = b − γ = T (A) − T (R )(1 − γ ) (1.107) g2 f 8π2 2 2 f f  f f with ∂ log Z (µ, µ ) γ (µ) = µ f 0 (1.108) f ∂µ

For the case of SU(Nc) with Rf = F ∀f = 1, ..., Nf we arrive ﬁnally at the following result g3 β (g ) = − c 3N − N 1 − 2γ (1.109) c c 16π2 c f

2. NVSZ scheme: exact beta function

This scheme is most similar to the one used for 1PI effective actions. It not only involves canonically normalized kinetic terms for the chiral fields, but also for the gauge fields. In components, the lagrangian (1.105) starts as follows 1 i L (µ) = − F a F µν + λa (σµ) Dabλ¯αb˙ + ... (1.110) c 4g(µ) µν a g2(µ) α αα˙ µ Hence the kinetic terms of gauge bosons and gauginos are not canonically normalized. “Canonical normalization”means that the coupling constant is not present in front of the

23 kinetic terms, and shows up in the covariant derivatives. This leads to the usual Feynman rules, with coupling constants associated to vertices, instead of propagators. In our case, one would naively think that canonical normalization can be easily achieved through the following rescaling redefinition V = gVc. Anticipating that this guess is wrong, we will replace it by another one V = gcVc (1.111) and try to find the relation between g and gc. Inocent as it looks, this change of variables is again anomalous. In fact, looking at (1.1) it is easy to recognize the action of (1.111) on the component fields a iβ a a iβ a ¯a α˙ −iβ¯ a α˙ a iβ a Aµ = e Ac µ , λα = e λc α , λ = e λc ,D = e Dc (1.112)

∗ with iβ = log gc(µ) = −iβ . This again looks like a (complexiﬁed) chiral tranformation in which the Weyl fermions transform with charge 1(though the parameter has been continued to the imaginary axis). Unfortunately the trick that worked before cannot be used here, as ther is no way to express the (complex) transformation of the gluinos as a complex chiral rotation of hte vector multiplet (which is real by deﬁnition). We simply quote the result for how the measure transforms

log gc (DV ) = (DgcVc) = (De Vc) T (A) Z = (DV ) exp log g2(µ) d4x(d2θTrW 2 + h.c.) (1.113) c 32π2 c Notice that this has the same form as an adjoint-valued chiral multiplet anomaly, but with the opposite sign, and iα = log gc. With all these ingredients let us proceed Z 1 Z 1 Z = (DV ) exp − d4yd2θ TrW 2(V ) + h.c. 4 g2(µ) Z 1 Z 1 = (Dg V ) exp − d4yd2θ TrW 2(g V ) + h.c. c c 4 g2(µ) c c Z 1 Z 1 T (A) = (DV ) exp − d4yd2θ − log g2(µ) TrW 2(g V ) + h.c. c 4 g2(µ) 16π2 c c c Canonical normalization will be achieved if the whole prefactor turns out to be exactlyt 2 gc (µ), or 1 1 T (A) 1 2 = 2 + 2 log 2 (1.114) gc (µ) g (µ) 16π gc (µ) This important equation, sometimes known as the Shifman-Veinstein equation, performs the change of variables gc = gc(g). With it one may compute Z(µ, µ0; gc), and using (1.106) 1 1 T (A) 1 2 = 2 + 2 log 2 (1.115) gc (µ) g (µ) 16π gc (µ) Now it is straightforward 1 T (A) 2 1 β 2 1 − 2 gc = β 2 (1.116) gc 16π g

24 and ﬁnally, from (1.107) obtain the NSVZ beta function.

3 P g 3T (A) − T (Rf )(1 − 2γf (gc)) β(g ) = − c f (1.117) c 32π2 g2T (A) 1 − c 16π2

Notice that, when we write the lagrangian canonically normalized, even for pure N = 1 SYM, the beta function receives corrections from all loops. Expanding (1.117) in power 2 series of gc should match all the higher loop perturbative calculations in a very particular scheme.

Discrete Chiral Symmetry.

The original ﬂavour symmetry of the UV action(1.50) is U(Nf ) × U(Nf ) × U(1)R ∼ SU(Nf ) × SU(Nf ) × U(1)B × U(1)A × U(1)R. The quantum numbers under these groups of the chiral and vector multiplets are given in table 1. In order to see how they arise iα remember that the chiral anomaly of a single left-handed Weyl fermion ψα → e ψα shifts the θ parameter by θ → θ − T (R)α (1.118) We expect a total shift of the θ parameter given by θ → θ − nα where n receives contributions from all the chiral fermions in the spectrum.

Notice that, since θ is an angular variable with period 2π, an exact discrete remnant Zn 2π (gauge) symmetry of any anomalous U(1) is given by α = m n , m = 1, 2, ..., n. Let us compute n in some cases.

2C(R) SU(Nf )L × SU(Nf )R U(1)B U(1)A U(1)R U(1)AF θ 0 0 0 0 1 1

Nf −Nc Qf ∼ φf 0 ( Nf , 1 ) 1 1 0 Nf

Nc ψf 1 ( Nf , 1 ) 1 1 -1 − Nf

f ˜f Nf −Nc Q˜ ∼ φ 0 ( 1 , N¯f ) -1 1 0 Nf

˜f Nc ψ 1 ( 1 , N¯f ) -1 1 -1 − Nf

V a 0 0 0 0 0 0

a a W ∼ λ 2Nc 0 0 0 1 1

n 0 2Nf 2(Nc − Nf ) 0

Table 1.

25 Nc From the last column observe that U(1)AF = U(1)R + (1 − )U(1)A is the anomaly free Nf combination of U(1)R and U(1)A, as is revealed by the zero in the last entry. In summary, anomalies break U(1)A ,→ Z2Nf and U(1)R ,→ Z2(Nc−Nf ), while there is an anomaly free combinations U(1)AF which remains unbroken.

1.4. Non Perturbative Superpotentials

1.4.1. Nf < Nc massless: The Aﬄeck-Dine-Seiberg Superpotential

The non perturbative induced superpotential must be a certain function of the massless de- f grees of freedom W = W(M g), constrained by the symmetries of the microscopic action. The only true (non-anomalous) symmetry U(1)AF is an R−symmetry, and hence W should transform with weight 2 under it. From table 1 we ee that detM ∼ det(QQ˜) transforms with AF charge 2/(Nc − Nf ). Correct scaling dimension 3 is achieved by adjoining the apropriate powers of Λ. The unique answer is given by the so called Affleck-Dine-Seiberg superpotential: Λb 1/(Nc−Nf ) W = (N − N ) (1.119) dyn c f det M where, remember, b = 3Nc − Nf . In view of the extension to more complicated examples we shall review an alternative and fully equivalente route here. Forget about U(1)AF , and turn all anomalous symmetries non-anomalous by endowing the parameters if the theory with compensating tranformation rules. In the case at hand, if θ changes as n θ → θ + nα ⇒ τ → τ + α (1.120) 2π this compensates for the anomalous jacobian 1.118. This is nothing but a trick, that handles anomalous symmetries as spontaneously broken symmetries, by regarding the parameters in the accion as vev’s of background fields in some more fundamental theory. After integrating out the massive fields, the low energy effective action still will parametrically depend on these background fields. Now the symmetries must be respected by the low energy effective superpotential, which, as a function includes the parameters. At low energy the dynamical scale Λ also gets transformed, through its link with the 4πi θ microscopic coupling constant τ = g2 + 2π . 2 − 8π +iθ Λb = µe2πiτ(µ) = µe g2 → Λbeinα (1.121) With all this we may build the following table

U(1)B U(1)A U(1)R dimension U(1)s f M g 0 2 0 2

detM 0 2Nf 0 2Nf b Λ 0 2Nf 2(Nc − Nf ) b

26 Table 2.

The explicit construction of the superpotential Weff has to cope with the following global symmetries:

f - SUL(Nf ) × SUR(Nf ). This enforces M g to enter through the combination det M.

b - U(1)A invariance constraints Weff (Λ, det M) = f(Λ / det M).

2 - U(1)R. Being an R symmetry, the invariant is d θWeff hence Weff must have charge 2. This yields Λb 1/(Nc−Nf ) W = C (1.122) eff det M

4 2 - U(1)scale. The invariant is now d xd θWeff , so Weff must transform with weight 3. This is true for (1.119)

b [Weff ] = ([Λ ] − [det M])/(Nc − Nf )

= (b − 2Nf )/(Nc − Nf ) = (3Nc − Nf − 2Nf )/(Nc − Nf ) = 3

The fact that scale invariance is automatic once U(1)R is safe is a manifestation of the anomaly multiplet structure, i.e. the fact that the energy momentum tensor and the R-current belong to the same supermultiplet.

Nf = 0 Chiral Symmetry Breaking through Gaugino Condensation

Expression (1.119) is kown as the Affleck-Dine-Seiberg superpotential. It is ill defined for Nf ≥ Nc. In the limit Nf → 0 we still may define the Affleck-Dine-Seiberg superpotential by setting det M = 1, hence using b = 3Nc

b 3 Wdyn = Nc Λ Nc = NcΛ (1.123)

This dynamically generated superpotential has two consequences

Gaugino Condensation Because gauginos of supersymmetric Yang Mills theories are massless and, at low energies, strongly interacting fermions, it is natural to ask whether we can reasonably expect pair condensation. This would be like quarks in QCD or like Cooper pairs in BCS theory of superconductivity. Once we have integrated out hte gluons, the eﬀective lagrangian written in terms of eﬀective degrees of freedom can help us in this direction Z a 1 a τ R d6x trW 2+h.c. hλ λ i = [dV ] λ λ e 16π a Z a Z 1 2 ( τ R d6x trW 2− h.c.) = [dV ] trW e 16πi Z(τ) θ=0 Z 1 ∂ τ R d6x tr W 2 = 16π [dV ] e 16πi Z(τ) ∂τ θ=0

27 ∂ = 16π (log Z[τ])| ∂τ θ=0 ∂ Z = 16πi d2θ W (τ) ∂τ eff ∂ = 16πN Λ3(τ) c ∂τ

∂ 3 2πiτ = 16πiN µ e Nc c ∂τ

= − 32π2 Λ3 (1.124)

Spontaneous Chiral Symmetry Breaking

The exact discrete remnant symmetry of the microscopic theory U(1)R ,→ Z2Nc (see paragraph after equation (1.207)) is spontaneously broken down to Z2 by the gaugino b b/Nc condensate. Indeed under U(1)R,Λ has charge 2Nc (see table 2), hence Λ has charge 2, 3 b 3 2Nc iα 3 2iα Λ = Λ Nc → Λ e Nc = Λ e (1.125) Therefore just α = 0, π leave the vacuum invariant. All the other elements α = m/Nc (m = 1, ..., Nc − 1) yield vacua which are diﬀerent but gauge equivalent to the one given in (1.124). In total we have Nc vacua.

0 < Nf < Nc Runaway Vacuum

f There is no supersymmetric vacuum for ﬁnite values of M g. The argument goes as follows: f ˜ g if hM gi 6= 0 then we expect quark condensates hψf ψ i 6= 0 that signal conﬁnement and spontaneous breaking of chiral symmetry. Up to here this is the same as for gluino f condensation. The point however is that ψf˜ψ is an F -term of the composite chiral operator ˜ g 5 Q Q . Hence F f 6= 0 signals supersymmetry breaking. Indeed f Q˜gQ

b 1/(Nc−Nf ) ∂Weff C Λ −1 g F f = = − M (1.126) M g f f ∂M g (Nc − Nf ) det M Hence the vacuum energy

X 2 X a 2 f E = |FM g | + |D | f,g a 2 b 2/(Nc−Nf ) C Λ −1 †−1 = 2 T r(M M ) (1.127) (Nc − Nf ) det M

f since we have chosen the Q˜g,Q conﬁgurations that solve the D ﬂatness conditions (1.18). f This forces a runaway behaviour M g → ∞ for a (supersymmetric) minimum at E = 0.

5 −1 −1 g f f use M δM = −M f δM g, with δ = ∂M g .

28 1.4.2. Nf < Nc mass deformation

Consider the mass term X f g Wtree = m gQ Q˜f . (1.128) fg Again, in order to look for a non-perturbative low energy superpotential we shall impose f invariance of Wtree by endowing m g with the required transformation properties. In this way we complete table 2

U(1)B U(1)A U(1)R dimension

detM 0 2Nf 0 2Nf b Λ 0 2Nf 2(Nc − Nf ) b g Q˜f Q 0 2 0 2 f m g 0 -2 2 1 trmM 0 0 2 3

det m 0 −2Nf 2Nf Nf

Table 3.

Now we have more freedom, and indeed the two combinations

1 b Λ Nc−Nf a = ; b = tr(mM) (1.129) det M have the right symmetries and dimensions.6 We may parametrize the solution as

f Weff (M g, Λ, m) = af(b/a) (1.130) 2 = a(c0 + c1b/a + c2(b/a) + ...) 1 1 b N −N Λ c f det M Nc−Nf = c + c TrmM + c (TrmM)2 + ... Nc,Nf det M 1 2 Λb

f In the limit m g → 0 we know the answer constains just the ﬁrst term. However, taking 8π2 − 2 also the weak coupling limit g(µ)|fixed µ → ∞ yields Λ = µe g b → 0, (b > 0). Hence the third and ongoing terms can survive in this double scaling limit. To avoid this we conclude that c2 = c3 = ... = 0, and then obtain the simple answer Weff = Wdyn + Wtree,

1 b Λ Nc−Nf W (Q, Q˜; m, Λ) = (N − N ) + Tr mM (1.131) eff c f det M

6 1/Nf Other combinations like (det m det M) seem pathological in the limit Nf → 0.

29 Vacuum Stabilization

There is no more runaway behavior.

1 b Nc−N ∂Weff Λ f −1 f f g = − M g + m g = 0 (1.132) ∂M f det M from here

1 b Λ Nc−Nf mf = M −1 f (1.133) g det M g

Nf −Nc N −N det m = Λb c f (det M) Nc−Nf

Nf Nc−Nf b Nc − det M = Λ (det m) Nc (1.134)

Inserting (1.134) back into (1.133) we arrive at

1 hM f i = Λb det m Nc m−1 f (1.135) g Nc,Nf g

th This formula displayes the expected Nc solutions through the (1/Nc) root.

What is the number of vacua?. In general the ﬂavor symmetry is broken by the mass term (1.128) as follows

SU(Nf )L × SU(Nf )R × U(1)B × U(1)B × U(1)A ,→ U(1)B

f f unless all masses are equal m g = mδ g in which case also the vectorial SU(Nf )V survives. Notice that although neither U(1)A nor U(1)R are classical symmetries, there is a combination U(1)R˜ = U(1)A + U(1)R that is. Actually it is equivalent to redefining the R-charge of the chiral superfields Q, Q˜ → 1 in Table 1 (the common value of R charge for the quiral superfields is fixed by the transformation of the superpotential, otherwise it is free). From the same table we observe that for this combination n = 2Nc and this signals, that the continuous classical symmetry is broken by instantons to the discrete quantum symmetry Z2Nc , as in the pure gauge theory. The elements of this discrete group are pha- 2π im f ses of the form e 2Nc , m = 1, ..., 2Nc. The chiral field condensate M g as charge 2 under 2mi 2π this discrete remnant symmetry, hence it transforms with e 2Nc which is left invariant only by the two elements m = 0,Nc. As in the flavourless case, this signals spontaneous symmetry breaking Z2Nc ,→ Z2 in as much as the truely distinct vacua are given by

f i(2m) 2π hM gi = c e 2Nc ; m = 1, ..., Nc with c as in the right hand side of (1.135). They are mapped to one another by the broken symmetry Z2Nc /Z2 given by m = 1, ..., Nc. Therefore we again obtain Nc vacua in complete agreement with index theorem calculations.

30 Gaugino Condensation in the Unbroken SU(Nc − Nf )

This is more or less obvious from the exponent 1/(Nc − Nf ) in (1.119). We can see it in two diﬀerent, but equivalent ways

f For light fields, i.e. m g → 0, we see that the vacuum condition (1.135) pushes the f N v.e.v.s hM gi → ∞. We have here a case of matching at theory scale µ f ∼ dethMi between a high energy with SU(Nc) theory with Nf flavours and a low energy theory pure gauge theory with SU(Nc − Nf ) flavours:

Λ3Nc−Nf Λ3(Nc−Nf ) = Nc (1.136) Nc−Nf (dethMi)

Dimensions match because det M has dimension 2Nf . Indeed, (1.136) is the only gauge and ﬂavour group invariant version of (1.79) that we have at hand. In terms of this the superpotential (1.119) reads

1 W = (N − N ) Λ3(Nc−Nf ) Nc−Nf = (N − N )Λ3 (1.137) c f Nc−Nf c f Nc−Nf

In view of (1.124) we expect a condensate

hλλi = 32π2Λ3 (1.138) Nc−Nf

f Take now values of m g >> 1 all ﬂavours become very massive with expectation values given by (1.135). We may integrate them out, and insert them in the Konishi anomaly relation (1.94)

1 hλλi = 32π2 Λ3Nc−Nf det m Nc (1.139) Nc,Nf

f µ=m g Again we are in the presence of a matching of theories (SU(Nc),Nf ) → (SU(Nc), 0) at scale det m, 3N −N Λ3Nc = Λ c f det m (1.140) Nc Nc,Nf Hence we ﬁnd again a gaugino condesate in the remaining pure gauge theory

2 3 hλλi = 32π ΛNc (1.141)

Holomorphic Decoupling

This is a powerfull tool to check the consistency of the Aﬄeck Dine Seiberg superpotential.

With the prefactor Nc − Nf the diﬀerent values of Nf chain together nicely. To see this, f Nf h let the matrix m g be dominated by m Nf = m >> m d ∼ 0. Then, instead of (1.131)

31 and (1.132) we ﬁnd

1 b ! Nc−N ΛN ,N f W (Λ ; m) ,→ (N − N ) c f + mM Nf (1.142) eff Nc,Nf c f det M Nf

1 b ! Λ Nc−Nf ∂Weff Nc,Nf −1 h h = − (M ) Nf = 0 (1.143) ∂M Nf det M 1 b ! Nc−N ∂W ΛN ,N f eff c f −1 Nf = − (M ) N + m = 0 (1.144) Nf f ∂M Nf det M

−1 h −1 From equation (1.143) we learn that (M ) Nf = 0. This means that M , and hence M, −1 Nf Nf are block diagonal matrices. Now, due to this fact we have that (M ) Nf ∼ 1/M Nf , ˜ Nf as well as det M ∼ det M · M Nf and, hence, from (1.144) we obtain

Nc−Nf 1   Nc−Nf +1  3Nc−Nf  Nc−Nf ΛN ,N 1 Nf  c f  M Nf =   (1.145)  det M˜ m

Inserting this back into (1.142) we get after a little algebra

1  3N −N  m Λ c f Nc−Nf +1 Nc,Nf Weff = (Nc − Nf + 1)   (1.146) det M˜

Invoking now the matching condition (1.79) with N˜c = Nc and N˜f = Nf − 1 gives

mΛ3Nc−Nf = Λ3Nc−Nf +1 (1.147) Nf ,Nc Nc,Nf −1 in the limit m → ∞ we obtain the correct decoupling, including the prefactor

1  3N −N −1  Λ c f Nc−Nf +1 m→∞ Nc,Nf −1 Weff (ΛNc,Nf ; m) ,→ Weff (ΛNc,Nf −1) = (Nc − Nf + 1)   (1.148) det M˜

1.4.3. Integrating Out and In

General Strategy for Nonperturbative Eﬀective Superpotentials

We will now try to put the previous exercise in a more general framework. The general setup deals with a supersymmetric ﬁeld theory and a treel level superpotential

X r i Wtree = grX (Φ ) (1.149) r where gr are classical sources for the gauge invariant functions Xr of the chiral superﬁelds i i i Q Φ = φ + ... transforming in the representation R of the gauge group G = s Gs, where each factor has an associated dynamical scale Λbs . The general procedure will be as follows

32 P a 2 1. Set Wtree = 0. We have a classical moduli space where V = s |D | = 0, The vev’s i hφ i 6= 0 break part or the full gauge symmetry. The gauge invariant functions Xr are natural coordinates that label inequivalent vaua. They are light ﬁelds in that the classical superpotential for them vanishes. If the Xr are constained classically we can add a lagrange multiplier to the eﬀective action.

2. Turn on gr and Λ (i.e. consider the full quantum theory. The full quantum wilsonian superpotential for the eﬀective theory is constrained by two kinematic constraints

r Holomorphy: Weff is a holomorphic function of the fields X and the couplingsgr. This last can be justified by thinking ofthe couplings as background vev’s of other superfields.

Symmetries: Weff is constrained by all the classical symmetries of the theory with Wtree = 0. Some of these symmetries can become anomalous for Λ 6= 0, or explicitely broken by Wtree. The selection rules are obtained by giving to Λ and gr compensating transformation rules. With them, an ansatz for Weff can be constructed. See for example (1.130) .

3. Explore the asymptotic behaviour. This includes weak coupling limit gr, Λ → 0, and i large vev’s hφ i >> 0. The key fact is that a holomorphic function, Weff in thi case, is determined by the asymptotic behavour and its singularities.

. At the end of the day, it is often the case that the resulting answer is of the form

r P r Weff (Λ, gr; X ) = Weff (gr =0) + r grX (1.150) r r = Wdyn(Λ; X ) + Wtree(gr; X )

In the word, it is linear in the sources gr. This is the case for the example (1.130). It is also the case in most of the examples known. When it is not, it is conjectured that there is r a redeﬁnition of X as funtions of gr such that (1.150) holds. This is the famous linearity principle.

Integrating Out

i Select one of the chiral ﬁelds Φˆ in the original theory, and let Xrˆ denote a subset of gauge invariant operators made out of Φ.ˆ Among them there is for example a mass term rˆ ˜ grˆX =m ˆ ΦˆΦ,ˆ or higher order invariants.

r rˆ r X r X rˆ Weff (Λ, gr, grˆ; X ,X ) = Wdyn(Λ; X ) + grX + grˆX (1.151) r rˆ

We may integrate out Xrˆ by solving their equations of motion

dWeff rˆ = 0r ˆ = 1, 2, ... (1.152) dX Xrˆ=hXrˆi

33 This is the same as demanding

∂W (Λ; Xr) dyn = −g (1.153) ∂Xrˆ rˆ

rˆ rˆ r From here we obtain hX i = hX i(Λ, gr, grˆ,X ), and inserting back into Weff we ﬁnd the eﬀective potential for the low energy theory below scalem ˆ .

r r rˆ W˜ eff (Λ, gr, grˆ; X ) = Weff (Λ, gr, grˆ; X , hX i) (1.154)

The “down”theory is no more linear in the couplings grˆ. However it is still linear in the gr

˜ rˆ rˆ rˆ dWeff ∂Weff (hX i) ∂hX i ∂Weff (hX i) r = + rˆ = X (1.155) dgr ∂gr ∂gr ∂X Therefore we write for it

r r X r W˜ eff (Λ, gr, grˆ; X ) = W˜ dyn(Λ, grˆ; X ) + grX (1.156) r

The eﬀective dynamical potential, contains now the additional couplings grˆ. The asymptotic behaviour dictates that ˜ r ˜ 0 ˜ r ˜ rˆ Wdyn(Λ, grˆ; X ) = Wdyn(Λ; X ) + WI (Λ, grˆ; X ) . (1.157)

r where W˜ dyn(Λ; X ) is the dynamical potential of the “down”theory. The term W˜ I vanishes in the limitm/g ˆ rˆ → ∞. This can be understood from the fact that the perturbation grˆ =m ˆ is gaussian, and hence, it can be integrated out exactly in the microscopic theory. The net effect can be absorbed into a redefinition of the dynamical scale through the usual matching condition ˜ ˜ Λ˜ b = Λbmˆ b−b ˜ 0 This is fully accounted for by the first term Wdyn in (1.157).

Integrating In

Although we got rid of the variables Xrˆ, we actually did not lose any information. This is very diﬀerent from the usual idea of integrating degrees of freedom, and is related to the linearity principle. In fact, W˜ , is simply a Legendre transform of W with respect to the variables Xrˆ. Therefore, in principle, it is possible to go back, and rescue the original variables by means of an inverse Legendre transform. To be precise, consider starting from rˆ the “down”theory Wdyn and perturbing by adding gauge singlets X at tree level. By the linearity principle, the full eﬀective action will look as follows

r rˆ r X r X rˆ W˜ eff (Λ˜, gr, grˆ; X ,X ) = Wdyn(Λ˜, grˆ; X ) + grX − grˆX (1.158) r rˆ

As usual, in this context we integrate out grˆ by demanding independence of the l.h.s.

dW˜ eff = 0 (1.159) dgrˆ hgrˆi

34 This equate to ∂W˜ dyn = Xrˆ (1.160) ∂grˆ

r rˆ This equation is to be solved for hgrˆi = hgrˆi(Λ, gr,X ,X ) one can show that the resulting expression is linear in gr (and, of course, independent of grˆ). By consistency it can be no other than the original perturbed action for the “up”theory

r rˆ r rˆ X r Weff (Λ˜, gr, hgrˆi; X ,X ) = Wdyn(Λ; X ,X ) + grX (1.161) r

Veneziano-Yankielowicz Glueball Eﬀective Superpotential

In all the previous reasonings, Λb appeared as a parameter that, in principle coul be b treated o equal footing with the gr. hence we may consider τ = log Λ as the source for the 1 2 composite gauge invariant chiarl superfield S = 16π2 trW . The effective ADS potential b 3 W˜ = NcΛ = NcΛ Nc serves to the purpose of computing the associated classical field

˜ b ∂Wdyn(Λ ) b 3 hSi = = Λ Nc = Λ (1.162) ∂ log Λb

b If we integrate this ﬁeld in, we are entitled to solve for hlog Λ (S)i = Nc log S and introduce it back to get

b b Weff (S) = W˜ dyn(hlog Λ i) − Shlog Λ i NC = NcS − S log S

= NcS(1 − log S)

From here the eﬀective action follows as

b b Weff (S;Λ ) = Wdyn(S) + log Λ S Λb = S log + Nc SNc S = N S 1 − log (1.163) c Λ3

However the meaning of this superpotential is unclear, since the chiral superﬁeld S is massive and our eﬀective actions are Wilsonian.

One could further integrate Nf ﬂavours in. The procedure is now clear: add Wtree = TrmM f and integrate out the parameters m g. Of course, to go to the “up”theory the scales have to match. Calling Λ˜˜b th scale-coupling that appears in (1.163), and Λb the one of the “upstairs”theory, we have clearly

Nf ˜ Y Λ˜ b = Λ˜ 3N−c−Nf = Λb det m. (1.164)

35 So, ﬁnally b f f Λ det m Weff (Λ, m g; S, M g) = S log + Nc − TrmM (1.165) SNc

Solving for dWeff /dm = 0 yields

SNf hmif = SM −1 f ; dethmi = (1.166) g g det M and hence we obtain for the “upstairs”dynamical potential

b f Λ Wdyn(Λ; S, M g) = S log + Nc − Nf (1.167) SNc−Nf det M

Because S is massive, it should be integrated out. After doing so, the Aﬄeck-Dine-Seiberg dynamical superpotential (1.119) is recovered.

1.4.4. Nf ≥ Nc: The quantum moduli space

No Dynamically Generated of Superpotential

b When Nc = Nf the U(1)R is anomaly free, ie Λ is inert. This makes it imposible to build a charge 2 superpotential in terms of Λ (at least for massless Q, Q˜). For Nf ≥ Nc the same expression as (1.119) could be used. However, the exponent 1/(Nc − Nf ) now is negative, and hence the dynamically generated scale Λb appears in the denominator. This makes it impossible to arise as an expression generated by instantons. As a consequence, contrarily to the case Nf < Nc, for Nf ≥ Nc nondynamical superpotential is generated and, therefore, a moduli space is still present in the quantum theory. Recall that the classical moduli space admits gauge invariant coordinates

f f M g = Q Q˜g

B[f1...fNc ] = Qf1i1 Qf2i2 ··· QfNc iNc (1.168) i1i2···iNc

˜ ˜ ˜ ˜ i1i2···iNc B[f ...f ] = Qf i Qf i ··· Qf i (1.169) 1 Nf 1 1 2 2 Nc Nc subject to algebraic constraints which are trivially satisﬁed in terms of the microscopic ﬁelds Q and Q˜. For Nc = Nf the quantum moduli space is a deformation of the classical one. For Nc = Nf + 1 quantum and classical moduli spaces are the same.

Nc = Nf : Quantum Deformed Moduli Space

f In this case there are Nf + 2 gauge invariant chiral composite superﬁelds M g ; B and B˜. The constraint f(M,B, B˜) = det M − BB˜ = 0 leads to a singular manifold at B = B˜ = 0, where f = df = 0. Seiberg proposed that quantum corrections modify this constraint in the following form det M − BB˜ = Λ2Nc (1.170)

36 The right hand side, being Λb is proportional to a one-instanton contribution. Observe f that, now, the origin M g = B = B˜ = 0 is no more in the moduli space. Hence quantum corrections have smoothed out the singularity. As a check of this conjecture we may decouple one ﬂavour and try to recover the known result for Nf = Nc − 1. To this aim, add the tree level mass term for the last ﬂavour Nf ˜ (Q , QNf ) Nf ˜ Nf Wtree = mQ QNf = mM Nf = Weff (1.171)

Nf since no additional piece is generated dynamically. For large m, the meson ﬁeld t = M Nf looses all its excitations (the kinetic term can be dropped) and it should be replaced by a function of the other ﬁelds that solves de quantum constraint. Writing

Nf X Nf +g Nf ˜ g det M = (−1) M g(det M Nf ) (1.172) g=1

˜ g Nf where by (det M Nf ) we mean the minor of the matrix element M g. Hence from the Nf constraint we can solve for M Nf and insert into (1.171) to ﬁnd

 Nf −1  m X W = Λ2Nc + BB˜ − (−1)Nf +gM Nf (detM˜ )g (1.173) eff ˜  g Nf  (det M) g=1

˜ ˜ Nf with (det M) = (det M Nf ). To find the vacuum manifold, we must solve the F flatness conditions. In particular for f, g = 1, ..., Nf − 1 we find

Nf −1 ˜ g ! ∂Weff X ∂ (det M) Nf = m(−1)Nf +gM Nf = 0 ∂M f g ∂M f ˜ h g=1 h (det M) ∂W B˜ eff = = 0 ∂B (det M˜ ) ∂W B eff = = 0 ∂B˜ (det M˜ ) (1.174)

N which is solved generically with M f g = B = B˜ = 0. Inserting this back into (1.173) only the ﬁrst term survives 2N 3N −N mΛ c mΛ c f Weff = = ˜ ˜ det M det M Nf =Nc ˜ b b mΛ Λ ˜ Nc,Nf Nc,Nf = = det M˜ det M˜ Nf =Nc ˜ Nf =Nc−1 ˜b Λ ˜ ˜ Nc,Nf = (Nc − Nf ) det M˜ ˜ Nf =Nc−1

37 hence the down superpotential comes out with the correct coeﬃcient. The constrained manifold allows for diﬀerent patterns of symmetry breaking depending on the vev’s of M,B and B˜. However in no point of moduli space is the full chiral symmetry broken. Indeed, for Nc = Nf and from table 1, we recognize that al scalar quarks are neutral under U(1)AF , hence this group will generically remain unbroken. Other points of interest show partial breaking of the global symmetry are

f 2 f M g = Λ δ g ,B = B˜ = 0. At this point we have a pattern of symmetry breaking SU(Nf )L × SU(Nf )R × U(1)b × U(1)AF → SU(Nf )V × U(1)B × U(1)AF

f Nc M g = 0 ,B = −B˜ = Λ , where the pattern of symmetry breaking now is SU(Nf )L × SU(Nf )R × U(1)B × U(1)AF → SU(Nf )L × SU(Nf )R × U(1)AF At either of these points we have a certain current algebra realized in terms of two theories, a microscopic (V, Q, Q˜) and a macroscopic (M,B, B˜). A strong test of the picture is provided by anomalous triangles, which can be computed in either of both theories, and must yield the same answer, as is indeed the case.

Nf = Nc + 1 2 f f ˜ In this situation we have Nf +2Nf gauge invariant polynomials M g,B and Bf satisfying the classical constraints

f f ¯ g −1 g ¯ g B¯f M g = M gB˜ = 0 ; det M(M )f = B¯f B˜ (1.175) where ¯ B¯ = B[f1...fNc ] ; B˜ = ff1...fNc B˜ ; (1.176) f ff1...fNc f [f1...fNc ] We cannot deform this constraint in a way covariant with the global symmetries in the absence of masses. Therefore the classical constraints remain the same. In fact the (1.35) can be thought of as the vacuum equations ∂W/∂M = ∂W/∂B = ∂W/∂B˜ = 0 coming from the following auxiliary superpotential 1 W = (B˜¯ M g B¯f − det M) (1.177) Λb g f This should not be thought of as a dynamically generated superpotential, but rather as a trick to envisage the constrained manifold (1.175) as the vacuum surface embedded in f a larger space where all ﬁelds M g,B and B˜ are physical and independent but couple through (1.177). As a check of this ansatz, let us decouple the last ﬂavour, by adding a mass term to (1.177) 1 ¯ W = (B˜ M g B¯f − det M) − mM Nf (1.178) Λb g f Nf

Nf i The F ﬂatness equations ∂W/∂M i = ∂W/∂M Nf = 0 with i < Nf reduce M to the following form ! M˜ 0 0 ¯ f 0 M = ; B¯ = ; B˜ = N (1.179) Nf f ¯ ¯ f 0 M Nf BNf B˜

38 ¯ ¯˜ ˜ so, renaming BNf = B and BNf = B, the superpotential takes the form 1 W = ( BB˜ − det M˜ ) − m M Nf (1.180) Λb Nf

Finally the equation of motion ∂W/∂M Nf = 0 implies, with Λb = Λ2Nc−1 Nf Nc,Nc+1 det M˜ − BB˜ = mΛ2Nc−1 = Λ2Nc (1.181) Nc,Nc+1 Nc,Nc and the correcto deformed moduli space for Nc = Nf is recovered. The point M = B = B˜ = 0 is now in the vacuum manifold. At this point the full global symmetry of the lagrangian is preserved. Hence we have conﬁnement without chiral symmetry breaking. At this point, anomalies provides a maximal set of nontrivial triangles to be matched.

Nf ≥ Nc + 2

The previous construction does not admit a further generalization to higher values of Nf − N − c ≥ 2. There is no way to write a superpotential with U(1)AF charge 2 that reproduces the classical constraints through the vacuum equations, and the anomaly matching conditions are not satisﬁed. A better idea is needed.

1.5. Seiberg Duality

Recall the original form of the barionic operators given in (1.27) and (1.28) and repeated below

B[f1...fNc ] = Qf1i1 ... QfNc iNc (1.182) i1...iNc B˜ = Q˜ ... Q˜ i1...iNc (1.183) [f1...fNc ] f1i1 fNc iNc

The antisymmetric set of indices Nc ≤ Nf indices signals that this is a bound state of i Nc objects (quarks (Q , Q˜i)) transforming in the fundamental of SU(Nc) gauge. Seiberg’s proposal starts by going over to the Hodge duals

1 B¯ = Qf1i1 Qf2i2 ··· QfNc iNc (1.184) [g1...g ˜ ] g1···g ˜ f1···fNc i1i2···iNc Nc Nc! Nc

[g ...g ] ¯ 1 N˜c 1 g ···g f ···f i i ···i B˜ = 1 N˜c 1 Nc Q˜ Q˜ ··· Q˜ 1 2 Nc (1.185) f1i1 f2i2 fNc iNc Nc! i with N˜c = Nf − Nc and interpret them as bound states of N˜c objects (dual quarks (qi, q˜ )) transforming in the fundamental of SU(N˜c).

i i ··· i ¯ 1 2 N˜c B[g ...g ] = qg1i1 ··· qg ˜ i ˜ (1.186) 1 N˜c Nc Nc

[g1...g ˜ ] ¯ Nc g1i1 g ˜ i ˜ B˜ = q ··· q Nc Nc i i ···i (1.187) 1 2 N˜c

39 The number of ﬂavours is the same Nf ≥ N˜c.

∗ SU(Nf )L × SU(Nf )R U(1)B U(1)A U(1)AF D(g =0) D(g =g )

N˜c 3 N˜c Qf ( Nf , 1 ) 1 1 1 Nf 2 Nf

f N˜c 3 N˜c Q˜ ( 1 , N¯f ) -1 1 1 Nf 2 Nf

f N˜c N˜c M g (Nf , N¯f ) 0 2 2 2 3 Nf Nf f ...f Nf ˜ ˜ 1 N˜c NcNc 3 NcNc B Nc Nc N Nc 2 N Nc f f N ˜ ˜ ˜ f NcNc 3 NcNc Bf1...f ˜ −Nc Nc N Nc 2 N Nc Nc f f

Table 3.

An important obervation is the following: gauge symmetries are related to a redundant description of physics. Duality, in the sense of universality says there is nothing that pre- vents having two such different UV descriptions that yield equivalent IR physics. However, global symmetries survive intact and must be the same all along the RG trajectory. The first thing we would try to do is to assign correct global charges to the dual microscopic fields. Since U(1) charges are additive, the correct way to assign quantum numbers to the g dual quarks qf , q˜ is to partition those of the baryon operators, among the new constituent fields. This leads to the first two lines in the following table.

∗ SU(Nf )L × SU(Nf )R U(1)B U(1)A U(1)AF D(g =0) D(g =g )

Nc Nc Nc 3 Nc qf (N¯f , 1) 1 N˜c N˜c Nf 2 Nf

f Nc Nc Nc 3 Nc q˜ (1 ,Nf ) - 1 N˜c N˜c Nf 2 Nf

f N˜c N˜c T g (Nf , N¯f ) 0 2 2 1 3 Nf Nf

g Nc Nc Nc M˜ f (Nf , N¯f ) 0 2 2 2 3 N˜c Nf Nf N ˜ ˜ ¯ f NcNc 3 NcNc Bg1...g ˜ Nc Nc N Nc 2 N Nc Nc f f g1...g ˜ N ˜ ˜ ¯˜ Nc f NcNc 3 NcNc B −Nc Nc N Nc 2 N Nc f f

Table 4.

f f It is fairly obvious from this table that the “magnetic”meson operator M˜ g =q ˜ qg is not f the counterpart of the “electric”meson operator Mg . For this reason, Seiberg introduced

40 f an additional fundamental ﬁeld, a color singlet T g transforming in the (Nf , N¯f ) represen- f tation, and endowed with the same quantum charges as the “electric”meson operator M g. f Now, if T g is a new fundamental chiral superﬁeld, there should be a U(1)T under wich only T is charged. This symmetry was not observed in the electric theory. Both problems, f f the absence of magnetic mesons M˜ g =q ˜ qg and U(1)T are solved by postulating the existence of a relevant superpotential

f gi W(q, q,˜ T ) = λ qfi T g q˜ = λtrT M˜ (1.188) where λ is a dimensionless constant. Observe that, interestingly enough, this expression has the right U(1)B × U(1)AF charge (0, 2) of a superpotential. However U(1)T is explicitely f broken as desired. Moreover, at a generic point in moduli space hT gi= 6 0, this potential gives mass to the magnetic meson operator, which hence decouples from the low energy eﬀective action. Next we face the following questions

1. In which sense is this duality a symmetry?

2. What are the consistency checks that it passes?

To answer the ﬁrst question we start looking carefully at the RG behaviour of the dual pair. Recall that the exact beta function was given by

g3 3N − N (1 − γ(g2)) β(g) = c f (1.189) 16π2 g2 1 − Nc 8π2 2 2 2 g Nc − 1 2 γ(g ) = − 2 + O(g ) 8π Nc where γ is the anomalous dimension of the mass. The structure of the numerator in (1.189) reveals a competition between the anti-screening of gluon, and the screening of matter fields. For different values of Nf and Nc, there is winning of either effect, or even an interesting window (conformal window) where they equilibrate. A carefull analysis of this exact beta function leads to the following phase diagram.

3Nc ≤ Nf Non Abelian IR Free Electric Phase.

In the last interval, 3Nc ≤ Nf , the theory loses asymptotic freedom and gains, instead, IR freedom. The potential among distant charges behaves at long distances as V (r) ∼ e2(r)/r with e2(r) = 1/ log(Λr). Much like QED, but with gluons (therefore the name of this phase). r = Λ−1 signals, instead, a Landau pole.

3 2 Nc < Nf < 3Nc Interacting Non-Abelian Coulomb Phase, ( conformal window ). In this interval, the coupling is presumed to approach a non trivial IR ﬁxed point ∗ g → g . This can be proven rigurously in a certain scaling limit where Nc,Nf → ∞. In general it is postulated that this is always the case in this interval. Therefore, the LEEA is a nontrivial superconformal ﬁeld theory. Because of conformal invariance, the potential among distant sources has to behave as V (r) ∼ 1/r for large r. That is why we refere to this as a non-Abelian Coulomb phase.

41 The full scaling dimension of a (composite) operator D(O) = D0(O)+γ(O) where D0 is the engineering dimension and γ the anomalous dimension. It is to be calculated 2 in perturbation theory as a series expansion γ = γ1g +.... Therefore the engineering dimension coincides with the full dimension in a free theory D0 = D(g = 0), i.e. at a trivial fixed point. The other interesting posibility arises in a superconformal field theory (for example, as is now the case, at a nontrivial fixed point g = g∗). The superconformal algebra has a nonanomalous U(1)R. From this algebra, the exact scaling dimension of chiral operators satisfies 3 D = |R| (1.190) 2

Back to our case, we identify U(1)R with our anomaly free combination U(1)AF . This allows us to complete the two final columns in figures 3 and 4. It is noteworthy to remark that the same values of γ can be obtained by demanding the vanishing of the numerator in (1.189). For example, since Q and Q˜ build up a single flavour ˜ β = 0 ⇒ γ(Q, Q˜) = −3 Nc + 1, and hence D(M) = 2 + γ = 3 Nc , and D(B) = Nf Nf Nc 3 N˜cNc Nc + γ = . 2 2 Nf

3 Nc + 1 < Nf ≤ 2 Nc Nonabelian IR Free Magnetic Phase. ∗ 3 As the Nf decreases past 3Nc, the fixed point g increases. For Nf ≤ 2 Nc is reached 3N˜c N˜ ≤ Nc/2 and the dimension D(M) = ≤ 1. This cannot correspond to a Nf unitary superconformal field theory, for which the smalles conformal dimension is 1. Therefore the theory must be in a different phase. From the point of view of the electric theory looks as standard asymptotically free theory, strongly coupled in the IR. However, from the point of view of the, as yet conjectured, dual magnetic theory 3 ˜ ˜ Nc + 1 < Nf ≤ 2 Nc corresponds to the inequality 4 + Nc ≤ 3Nc ≤ Nf . Therefore the magnetic theory is in a Non-Abelian IR Free Phase. The field content is therefore best f analized in the IR trivial fixed point, and consists of quarks qf , q˜ and the additional f field T g. Still we have to think about the Yukawa type superpotential (1.188). The dimension of this operator, which is 3 at the IR trivial fixed point where T, q and q˜ are free, has to be computed in perturbation theory. It turns out to be irrelevant and therefore vanishing in the IR limit. Therefore the magnetic theory looks around energy scales µ → 0 pretty much like the electric theory around energy scales µ → ∞.

1.5.1. Duality in the Conformal Window

˜ 3 Under Nc → Nc = Nf − Nc, the band 2 Nc < Nf < 3Nc maps over to the same interval 3 ˜ ˜ ˜ 2 Nc < Nf < 3Nc. The extrema are exchanged and the fixed point is at Nf = 2Nc = 2Nc. Therefore both theories, in the IR are in a Non-Abelian Coulomb Phase. In the IR, the electric meson field M and the magnetic singlet T describe the same macroscopic field, since their dimensions match. Actually we are speaking about three different fixed points. At λ = 0 we have a fixed point at g∗ where the singlet field T is free (it only has kinetic term). Therefore, at this point the dimension of the perturbation (1.188) is equal 1 + 2 3 Nc ≤ 3. 2 Nf

42 Hence it is a relevant perturbation, and it drives the theory in the IR to another fixed point (g∗, λ∗). It is this fixed point the one that describes the same physics as the electric theory. In particular, T and M flow to the same IR meson operator since both their quantum numbers and dimensions match. At the UV fixed point, however, there is not such a simple matching as with the barions. The UV dimensions of T and M disagree by an amount of one. In order to relate them we must introduce a scale µ 1 1 T f = M f ≡ Qf Q˜ (1.191) g µ g µ g An independent way of introducing this independent scale is as the crossover scale of both asymptotically free theories whose dynamical scales are Λ and Λ.˜

˜ ˜ Λ3Nc−Nf Λ˜ 3Nc−Nf = (−1)Nc µNf (1.192)

The behaviour of the RG ﬂows embodied in this equation shows that the gauge coupling of one theory becomes weaker as the one of the dual becomes stronger.

Duality is an involution

The dual of SU(N˜c) and Nf ﬂavours, contains again Nf ﬂavours in the fundamental of ˜ the gauge group SU(N˜ c) = SU(Nc). The relation (1.192) does not transform correctly. A second duality trasformation brings it to

˜ ˜ ˜ 3N c−Nf ˜ Λ˜ 3Nc−Nf Λ˜ = (−1)N c µ˜Nf (1.193)

Since the ﬁnal theory and the original gauge group are the same so are they scales and therefore (1.193) is the same as

˜ Λ˜ 3Nc−Nf Λ3Nc−Nf = (−1)Nc µ˜Nf (1.194) which, compared with (1.192) shows thatµ ˜ = −µ. Since the dual of the dual barions are the original ones, we expect the constituent quarks f f to be none other than the original (Q , Q˜f ). In addition there is a gauge singlet U g that plays the role of the magnetic meson M (m), and shoud be identiﬁed with it (in the free UV) through a relation analogous to (1.191) 1 1 U f = M˜ f = qf q˜ (1.195) g µ˜ g µ˜ g The picture completes with the addition of the corresponding superpotential that adds up to the one written in (1.188), but expressed in the original variables through (1.191)

f g f g W = qf T gq˜ + Q˜f U gQ 1 1 = trMT + Q˜ M˜ f Qg µ µ˜ f g 1 = trMM˜ − Q˜ M˜ f Qg (1.196) µ f g

43 The ﬁrst term is a mass term for both M and T which therefore may be integrated out through their equations fo motion.

∂W ˜ g f = M f = 0 ∂M g ∂W = M g − QgQ˜ = 0 (1.197) f f f ∂M˜ g

Hence, with the advised equationµ ˜ = −µ we obtain the desired relation M = QQ˜ which makes the whole picture consistent.

Consistency with Deformation

1 2 v.e.v. deformation Turn on a v.e.v. hM 1i = a . For large a, the eﬀect on the electric theory is tu Higgs the gauge group SU(Nc) ,→ SU(Nc − 1). mass deformation

1.6. Conifold Field Theory (Klebanov-Witten)

1.6.1. Free Theory

Let N+ stand short for N + M. Consider an N = 1 QFT with U(N+) × U(N) gauge symmetry, with (Af ,Bf ) chiral superﬁelds f = 1, 2. Hence this case falls in the general case analyzed before, where (A1,B1) and (A2,B2) build up two ﬂavors (Qf , Q˜f ), f = 1, 2 = Nf . The gauge group is a direct product. Otherwise one should think of it as a unitary group with rank two Kronecker product matrices. ( T A = tA ⊗ 1 A = 1, ...N 2 A A (N+) (N+) + g ∼ 1 + αAT T A A 2 (1.198) T(N) = 1 ⊗ t(N) A = 1, ..., N

For tA and tA we may choose the standard basis (N+) (N)

(tmn )p = δmpδn m, n, p, q... = 1, ...., N (N+) q q + ab c ac b (t(N)) d = δ δ d a, b, c, d, ... = 1, ...., N (1.199)

Otherwise indices i, j, ... haver their rank restricted by the context. Correspondingly, ma nb Af ,Bg are rectangular matrices in color space. As representations of U(N+) × U(N) the chiral superﬁelds transform as

ma ¯ Af → (N+, N) nb ¯ m, n = 1, ..., N+, a, b = 1, ..., N Bg → (N+,N)

τ Z 1 Z t L = d2θ W 2 + d2θd2θ¯ Af†e2V A + Bg†e−V B + ζV (1.200) 16πi 4 f g

44 Vacuum solutions are given by the D ﬂatness condition

N + 2 X A 2 X f† A ma nb f† A t ma nb |D | = Ama(t ⊗ 1) nbAf + Bma((−t ) ⊗ 1) nbBf A A=1 N 2 X f† A t ma nb f† A ma nb + Ama(1 ⊗ (−t ) ) nbAf + Bma (1 ⊗ t ) nb Bf A=1

N+ N 2 2 X f∗ na f∗ na X f∗ mb f∗ mb = AmaAf − BmaBf + −AmaAf + BmaBf m,n=1 a,b=1

mn i mi n where use of the standar basis for the Lie algebra u(N), (t ) j = δ δ j has been made. Performing a SU(N+) × SU(N) rotation be may go to a diagonal basis

ma (a) ma nb (b) nb Af = af δ ; Bf = bf δ (1.201)

(a) (b) where (af , bf ) represent 4N complex parameters. We arrive at

N 2 2 X A 2 X X (a) 2 (a) 2 |D | = 2 |af | − |bf | = 0 (1.202) A=1 a f=1 This imposes the vacuum conditions

(r) 2 (r) 2 (r) 2 (r) 2 |a1 | + |a2 | − |b1 | − |b2 | = 0 r = 1, ..., N (1.203)

For generic diagonal vevs. like in (1.201) there is an unbroken U(1)N symmetry. The unbroken generators are the diagonally embedded Cartan subalgebra tpp ⊗ 1 + 1 ⊗ (N+) pp pp i pi p i t(N) with (t )j = δ δ j = δ j.

h pp ma nb pp t ma nb (t ⊗ 1) nbAf + ((−t ) ⊗ 1) nbBf

pp t ma nb pp ma nbi + (1 ⊗ (−t ) ) nbAf + (1 ⊗ t ) nb Bf h (p) (p) (p) (p)i = af − bf − af + af = 0

(r) (r) When all eigenvalues are equal af = af , bf = bf all diagonally embedded generators [tA ⊗ 1 + 1 ⊗ tA ] ,A = 1, ..., N 2 are unbroken, and the gauge symmetry is (N+) (N) enhanced to U(N)V .

1.6.2. Adding a Superpotential

Let us add the following SU(2) × SU(2) invariant tree level chiral interaction

dg fh W = λ tr(AdBf AgBh)

= 2λ tr(A1B1A2B2 − A1B2A2B1) (1.204)

45 This represents a non-renormalizable interaction of mass dimension -4. Hence the coupling constant λ has mass dimension 1. The F−term vacuum conditions read

∂F ij = 0 → B1AgB2 − B2AgB1 = 0 ∂Ad (1.205) ∂F kl = 0 → A1BhA2 − A2BhA1 = 0 ∂Bf

1.6.3. Conifold Connection

The classical field theory is well aware that it represents branes moving on a conifold. g Consider the diagonal form given in (1.201) for Af and B . The F term equations (1.205) (r) (r) are trivially satisfied. Constraining af and bg by means of equation (1.203) cuts from 8N down to 7N real variables. The diagonal action maximal abelian subalgebra of dimension N (so far we are having U(N) instead of SU(N)) allows to mod out N phases. Hence 6N real or 3N complex variables remain. With them we can define 4N new complex numbers (r) wA via 1 W (r) = a(r)b(r) = √ w(r)σA (A = 1, ..., 4) fg f g 2 A fg constrained by the (in terms of af , bg trivially satisfied) equation

4 (r) X (r) det W = 0 = (w )2 (1.206) f,g fg A A=1

(r) which is exactly the conifold equation. Therefore the Wfg are coordinates that represent the position of the r0th D3-brane on the conifold.

1.6.4. Global Symmetries

With λ = 0 the model (1.200) enjoys the usual flavour U(2) × U(2) ∼ SU(2)L × SU(2)R × U(1)B × U(1)A symmetry. the SU(2) rotate the flavour indices Af and Bg. As usual, U(1)A and U(1)R are anomalous. The two gauge groups have different beta functions b = 3Nc − Nf (notice that (Af ,Bf ) build up 2 (Dirac) flavours in total)

SU(N+) SU(N) b = b = 3N+ − 2N ; b = ˜b = 3N − 2N+ ˜ Correspondingly we have two diﬀerent strong scales ΛSU(N+) = Λ and ΛSU(N) = Λ. Consider the following table

46 SU(N+) SU(N) SU(2) × SU(2) U(1)B U(1)A U(1)R

A N N¯ (2, 1) 1 1 1 f + 2N+N 2N+N 2

B N¯ N (1, 2) −1 1 1 f + 2N+N 2N+N 2

Λb 0 2 2M N+

˜˜b 2 Λ 0 N −2M

λ 0 − 2 0 N+N

The charges of Λ, Λ¯ and λ are ﬁxed by the conventional choice made for the charges of Af and Bf . For example

the U(1)R charges of have been selected to have a charge two superpotential with inert λ. The barionic and axial charges are conventional.

The U(1)A and U(1)R charges of Λ and Λ¯ are obtained as follows. Remember that the chiral iα anomaly of a single left-handed Weyl fermion ψα → e ψα shifts the θ parameter by 1 R = N or N¯ θ → θ − T (R)α T (R) = c c (1.207) 2Nc R = adjoint

We expect a total shift of the θ parameter given by θ → θ−nα where n receives contributions from all the chiral fermions in the spectrum. To restore the symmetry one may endow θ with compensating transformations properties 4πi θ n θ → θ + nα ⇒ τ = + → τ + α (1.208) g2 2π 2π The dynamical scale Λ also gets transformed

2 b 2πiτ(µ) − 8π +iθ b inα Λ = µe = µe g2 → Λ e (1.209)

ma ma From the point of view of SU(N+), the ﬁelds Af ,Bf contain 4N (indices f = 1, 2; a = 1 1, ..., N)) weyl quarks qA, qB with charge in the fundamental (index m = 1, ..., N+). 2N+N b 2 This yields the U(1)A charge of Λ as . The U(1)R charge of chiral quarks qA, qB is N+ 1 1 α 2 − 1 = − 2 . Again there are 4N of them. But now we must count the gauginos λ with 1 charge 1 in the adjoint of SU(N+), and they contribute 2N+. Altogether 2N+ − 2 4N = 2M. The analysis for SU(N) is the same.

From the list of charges above we notice that U(1)R transformations with phase

2πi m e 2M ∈ Z2M ; m = 1, 2, ...., 2M

πi m b ˜b ±2πi m 2M aﬀect (Af ,Bg) → e 2M (Af ,Bg) whereas Λ and Λ˜ change by e 2M = 1, ie Λ, Λ˜ and λ are unchanged. This is the anomaly free Z2M remant of U(1)R. For M = 0 the full

47 U(1)R is anomaly free. Ultimately this justiﬁes the choice of charges 1/2 for the chiral multiplets.

Apart from the superpotential

dg fh W = λ tr(AdBf AgBh) there are other combinations of parameters and ﬁelds which are invariant under the global symmetries. For example

˜ Λ˜ b h i2M I = λ3M tr(A B A B )dgfh Λb d f g h Λ˜˜b = λM W 2M (1.210) Λb Another possibility comes from quotients of the form

dg fh (1) tr(AdBf )tr(AgBh) R = dg fh (1.211) tr(AdBf AgBh) with equal number of ﬁelds A and B in numerator and denominator but diﬀerently con- tracted ´ındices. There i salso an important invariant involving just scalar parameters

˜ J = λN++N ΛbΛ˜ b. (1.212)

Its logarithm plays the analog of the dimensionless coupling constant τ in N = 4 SYM. As a general rule, the tree level superpotential will be renormalized to take the following form dg fh (s) Weff ,→ λ tr(AdBf AgBh) F (I, J, R ) (1.213) for a, to be determined function F .

48 Cap´ıtulo 2

Selected Topics in Gauge Theories

2.1. Anomalies in Gauge Theories

Classical symmetries are symmetries of the Lagrangian. Quantum symmetries are symmetries of the path integral. Whenever a classical symmetry is not a quantum symmetry it is termed anomalous. The prototypical example is the path integral of a chiarl Weyl fermion ψ coupled to a gauge ﬁeld Aa. Z 4 † µ a a S[ψ, Aµ] = d x i ψ σ¯ (∂µ + iAµT )ψ (2.1)

Apart from the fact that there is a non-abelian gauge symmetry that involves transforming a iα Aµ, there also exist a global chiral U(1) symmetry whereby ψ → e ψ gets rotated by a global phase. An clever way to capture the Noether current is to make a local transformation, α → α(x), instead of a global one. This is not a symmetry, hence he action changes by Z iα(x) 4 † µ S[e ψ, Aµ] = S[ψ, Aµ] + i d x(iα(x))∂µ(ψ σ¯ ψ) . (2.2)

Not for constant α this must be a symmetry, hence the additional term must vanish, or µ ∂µjA = 0 with µ † µ jA = iψ σ¯ ψ (2.3) The Fujikawa method allows to compute the violation of the conservation equation, so that quantum mechanically it is replaced by 1 ∂ jµ = F aµνF a (2.4) µ A 32π2 µν The easiest way is to regard the path integral as a function of the background gauge ﬁeld

Z Z iα † −S[ψ,Aµ] iα −iα † −S[e ψ,Aµ] Z[Aµ] = DψDψ e = D(e ψ)D(e ψ )e (2.5)

Again, letting α(x) be local, we expect that Z Z iα(x) −iα(x) † −S[eiα(x)ψ,A ] † −S[ψ,A ]−R d4xiα(x)∂ jµ D(e ψ)D(e ψ )e µ = DψDψ J[α(x)]e µ µ A

49 Z † −S[ψ,A ]−R d4xiα(x)(∂ jµ −A(x)) = DψDψ e µ µ A where A(x) = exp[log J(x)] and J(x) is a Jacobian determinant associated to the change of measure. Since Z is independent of α(x) we may take functional derivative to obtain the new conservation law

δZ µ = h∂µj (x) − A(x)i = 0 (2.6) δiα(x) α=0 where the brackets stand for expectation values. A carefull evaluation of A(x) was ﬁrst done by Fujikawa, giving the following result 1 A(x) = TrF µνF˜ (2.7) 16π2 µν

50 Cap´ıtulo 3

The Minimal Supersymmetric Standard Model

3.1. Introduction

If supersymmetry was a real symmetry of the world, the spectrum of particles should come in representations of the algebra. In these representations, fermions and bosons are grouped together with equal masses. In 1976-77 Pierre Fayet made a ﬁrst attempt in this direction. In a sense this program resembles the one followed by P.A.M. Dirac when postulating that the proton would be the anti-particle of the electron. So, for example the photon γ and the neutrino ν, seem to be suited to be inside the same vector multiplet. Other possible susy pairs would be such as (W ±, e±). After realising that the program was failure, the standard point of view now is to accept that the susy partners have never been observed, and therefore, SUSY, if there, must be broken at the energy scale of 1 TeV. 3.1.0.1 The hierarchy problem

In high energy physics there are at least two fundamental scales: the Planck mass MP l ∼ 19 2 10 GeV deﬁning the scale of quantum gravity, and the electroweak scale Mew ∼ 10 GeV, deﬁning the electroweak symmetry breaking scale. The obvious questions to solve are

Why Mew << MP l ? This is the hiearchy problem Even if this hierarchy was postulated classically, is it stable? This is the ”naturalness”problem

To understand the naturalness problem, let us review the fate of masses in the Standard model. µ Gauge particles are massless because of gauge invariance. A term such as mAAµA is not invariant under Aµ → Aµ + ∂µα. Fermions are also massless in the Standard model because of of chirality. This means that ¯ mψLψR is not gauge invariant because tipically ψL and ψR live in diﬀerent (conjugate)

51 representations of the gauge group. The receive masses through the Higgs mechanism (e.g. H · ψψ gives mass to ψ after H gets a v.e.v.). 2 ¯ The Higgs, as any scalar particle can have a mass term mH HH which is not forbidden by any symmetry. Moreover, loop corrections are quadratic, hence they will shift the value of mH up to the cutoﬀ scale Λ of the theory. This would ruin the classical hierarcy mH ∼ Mew << MP l. Supersymmetry solves the naturalness problem by adding partners that, when running in the loops, cancel the quadratic divergences. This is the non-renormalization theorem at work. Hence supersymmetry does not explain the hierarchy, but once given, stabilizes it.

3.2. The MSSM

Matter in the standard model is chiral. This means that L and R chiralities transform under diﬀerent representations of the gauge group. It is customary to list only left handed chiralities. SU(3)×SU(2)×U(1) ψ U Q = (3 , 2 , 1/3) (q , q˜ ,F ) D L L qR N L = (1 , 2 , -1) (l , ˜l ,F ) E L R lR U c (3¯ , 1 , -4/3) (uc , u˜∗ ,F ∗ ) L R uR Dc (3¯ , 1 , 2/3) (dc , d˜∗ ,F ∗ ) L R dR Ec (1 , 1 , 2) (ec , e˜∗ ,F ∗ ) L R eR

3.2.1. Field content

The vector multiplets include now new fermions: gauginos and higgsinos. ± ± ± ± W = (AW , λW ,D ) (3.1) 0 0 0 0 W = (AW , λW ,D ) (3.2) A = (A, λ, D) (3.3)

This puts in danger the delicate cancellation of gauge anomalies that occurs in the Stan- dard Model. Gauginos pose no problem as they couple vectorially

3 3 a single Higgsino running in a triangle loop contributes proportionally to YH = +1 to the gauge anomaly. A solution is to introduce a second Higgs doublet of opposite hypercharge such that Y 3 + H1 Y 3 = (+1)3 + (−1)3 = 0. H2 Also this is required by the supersymmetric version of the Higgs mechanism. In this mechanism, the starting point is a massless vector supermultiplet and a chiral massless supermultiplet

52 Massless vector Massless chiral

V → (A⊥1,2 ; λ1,2) H → (h0 + ih1 , ψh1,2 )

After the Higgs mechanism, one of the scalars, say h0 provide the third longitudinal d.o.f for the vector potential. A massive supermultiplet has, in addition, four spin 1/2 d.o.f and an additional scalar

Massive vector

V → (A⊥1,2 ,Ak ∼ h0 ; λ1,2 , χ1,2 ∼ ψh1,2 , φ ∼ h1)

In summary, for each vector supermultiplet we need 2 bosonic degrees of freedom. Then ± 0 (W ,Z ) needs 6 bosonic degrees of freedoms. In a SU(2) doublet Hi, i = 1, 2 there are only 4. Let us introduce therefore a second doublet

SU(3)× SU(2)×U(1) 0 H1 H1 = − (1 , 2 , 1) H1 + H2 H2 = 0 (1 , 2 , -1) H2 such that, after giving mass to all vector bosons, still 2 degrees of freedom are left over. One is a scalar h0, and the other one a pseudoescalar A0. This is the MSSM content.

3.2.2. Couplings

The superpotential contains the relevant couplings. On one hand we need the Higgs to break SU(2) × U(1)Y → U(1)em. We can write bilinear couplings

WH = −µH1 · H2 = −µijHiHj (3.4)

µ has mass dimension 1, hence its smallness, as compared with other scales like uniﬁcaton scale or Planck scale should be given a good reason (µ problem). Cubic terms are needed to give Yukawa masses to the particles of the Standard Model

c c c WY uk = λdQ · H1D + λuQ · H2U + λeL · H1E (3.5) with color indices suppressed. We shall assume that the only scalars that acquire v.e.v. are the (electrically) neutral 0 Higgses H1,2, i.e. 0 hH1 i 0 hH1i = ; hH2i = 0 (3.6) 0 hH1 i then 0 c 0 c 0 c WY uk = −λdhH1 iDLDL + λuhH2 iULUL − λehH1 iELEL (3.7)

53 From here, the lagrangian in components has terms

1 X ∂2W L = − ψ¯c ψ + h.c. + ... 2 ∂φ ∂φ i,L j,L ij i j = λ hH0iψ¯c ψ + λ hH0iψ¯c ψ + λ hH0iψ¯c ψ + ... (3.8) d 1 dL dL u 2 uL uL e 1 eL eL

Notice that hypercharge conservation demands H2 to give mass to up quarks ψu, since we ∗ cannot make use of H1 . This is another reason to introduce two Higgs doublets.

3.2.3. R-parity

3.2.3.1 Dangerous couplings In the Standard Model, gauge invariance plus renormalizability restricts the Yukawa couplings to be ones in the previous section. In the MSSM this is not so, as there are more particles involved. For example, the following is a set of possible gauge invariant renormalizable couplings, L · LEc , Q · LDc and U cDcDc. They are dangerous because the violate lepton or baryon number.

L · LEc Q · LDc U cDcDc U1(B) 0 + 0 + 0 = 0 1 + 0 - 1 = 0 -1-1-1 = - 3 U1(L) 1 + 1 - 1 = 1 0 + 1 + 0 = 1 0 + 0 + 0 = 0

An important consequence would be the decay of the proton, through the Feynman diagram shown in ﬁg. 3.1

d e- d p+ u uc π0 uu

Figura 3.1: Proton decay mediate through the terms U cDcDc and Q · LDc in the superpotential.

Since we now experimentally that the proton lifetime is very long τ > 1025 years, this poses 15 a bound on the mass of the intermediary particle md˜ ≥ 10 GeV. For trilinear couplings of O(1) this implies that supersymmetry is so badly broken it will be useless as a possible solution to the hierarchy problem. We shall assume they are absent in the MSSM. But a more reﬁned way to rule them out is to postulate a new symmetry that forbids them. It must be a symmetry that treats

54 diﬀerently the SM particles from theis SUSY partners. Hence it better be a subgroup of the R-symmetry called R-parity. Let r be such that

iπ πR : θ → e θ = − θ ⇒ R(θ) = −1 (3.9) we say that θ is parity odd. Let us assign deﬁnite parities to the matter multiplets

R(Q) = R(L) = R(U) = R(D) = R(E) = −1

R(H1) = R(H2) = 1 (3.10) then c c c R(λdQ · H1D ) = R(λuQ · H2U ) = R(λeL · H1E ) = +1 (3.11) are allowed whereas

R(L · LEc) = R(Q · LDc) = R(U cDcDc) = −1 (3.12)

3.2.3.2 Experimental consequences There is a compact way to assign R parity to component ﬁelds. Given its baryon number is B, lepton number is L and spin is s,

R = eiπ(3B+L+2s) (3.13)

This implies that all susy partners have odd parity

BLS 3B+2L+S R q 1/3 0 1/2 2 1 q˜ 1/3 0 0 1 -1 l 0 1 1/2 2 1 ˜l 0 1 0 1 -1

Concerning experimental consequences we observe that:

Supersymmetric particles have to come in pairs, since +1 = (−1)(−1).

Among al the supersymmetric partners, there must be one that is the lightest (LSP). Having R = -1 it cannot decay anymore and therefore it must be stable. If the LSP is neutral (neutralino, higgsino, photino), then it is a weakly interacting massive particle (WIMP), and is a candidate for dark matter.

3.3. Electroweak symmetry breaking

In order to discuss gauge symmetry breaking we are entitled to analyze the full scalar potential. Now, in contrast to the SM, lots of scalar ﬁelds could acquire a v.e.v.. In order not to break color or R-symmetry we will set them all to vanishing v.e.v. except for the Higgs ﬁelds. Hence, the scalar potential is

i i V (H1,H2) = VF + VD (3.14)

55 For example

2 2! X i 2 i 2 X ∂W ∂W VF = |F | + |F | = + 1 2 ∂Hi ∂Hi i i 1 2 X 2 j∗ k = |µ| ijH2 ikH2 + (1 ↔ 2) ij 2 † † = |µ| H1H1 + H2H2 (3.15)

The D term potential involves contributions from both the SU(2) and the U(1)Y gauge vector supermultiplets

1 ~τ ~τ 2 1 g0 2 2 V = g2 H† H + H† H + −H†H + H†H . (3.16) D 2 1 2 1 2 2 2 2 2 1 1 2 2

0 where g and g /2 are, respectively, the couplings of SU(2) and U(1)Y . The quadratic contribution (3.15) is positive deﬁnite, and the quartic couplings in (3.16) are of order ∼ g2 hence small. Hence, if at all there is SSB, it would happen at a very 2 2 2 small scale MW ∼ g v . The crucial assumption is that spontaneous SUSY breaking induces new terms that trigger gauge SSB at a desired pattern. These include masses for scalar particles, as well as bilinear couplings 2 † 2 † VSB = mH1 H1H1 + mH2 H2H2 + (BµH1 · H2 + h.c.) . (3.17) ± One can show that the minimization of VF + VD + VSB happens for hH1,2i = 0 which implies U(1)em will not be broken. Therefore we can restrict ourselves to the neutral higgs components.

0 0 2 2 0 2 2 2 0 2 0 0 0∗ 0∗ VSB(H1 ,H2 ) = m1H1 |H1 | + m2H2 |H2 | + Bµ(H1 H2 + H1 H2 ) g2 + g02 + (|H0|2 − |H0|2)2 (3.18) 8 1 2 were we have introduced the mass parameter

2 2 2 m1 = mH1 + |µ| 2 2 2 m2 = mH2 + |µ| (3.19)

These parameters must be constrained by some physical requirements.

Stability 2 2 m1 + m2 > 2|Bµ|

0 0 Gauge symmetry breaking ⇒ H1 = H2 = 0 should not be a minimum

2 2 2 m1m2 < |Bµ| .

56 2 2 These two conditions imply that necessarily m1 6= m2. Deﬁning 0 0 v1 = hH1 i ; v2 = hH2 i the relevant magnitudes are

2 2 2 v2 v = v1 + v2 ; tan β = v1 in terms of which, the physical masses of the gauge bosons can be expressed as

g2 m2 − m2 tan2 β M 2 = (v2 + v2); M 2 = 2 1 2 (3.20) W 2 1 2 Z tan2 β − 1

3.4. Supersymmetry Breaking

If SUSY is true, it must be broken. The mass of the lightest squark must be well above that of the heaviest quark in the SM. However if we supersymmetry is spontaneously borken inside the MSSM, we have the condition

2 F 2 2 2 StrM = Tr(−1) M = TrMscalar − TrMfermions = 0 . (3.21) Hence, in mean, scalars are as light as fermions. The way out of this concundrum, is to postulate the existence of a hidden sector where the SUSY SSB occurs. The particles in this sector couple very weakly to those of the SM. Hence on one side they are diﬃcult to rule out and, on the other, the eﬀects of the SUSY breaking is weakly mediated to the observable sector. There are three basic scenarios for this mediation

Gravitational mediation.

Gravitons couple both sectors, and therefore, loop eﬀects are suppressed by 1/MP lanck ∼ 10−18GeV. In order to obtain a mass splitting, the amount of susy breaking we need, by mere dimensional analysis will be

M 2 ∆m = susy ∼ 1 TeV = 103GeV (3.22) MP lanck

√ 10 hence Msusy ∼ ∆m MP lanck ∼ 10 GeV. Gauge mediation

G = (SU(3) × SU(2) × U(1)) × Gsusy ≡ G0 × Gsusy

Matter ﬁelds are charged under both G0 and Gsusy which gives a Msusy of order ∆m i.e.E O(1) TeV. In that case, the gravitino mass m3/2 is given by

2 Msusy −3 m3/2 ∼ ∼ 10 eV . MP lanck

57 Anomaly mediation This is mediated by some auxiliary fields of supergravity. Effects are suppressed by loop counting. 3.4.0.3 Soft Susy Breaking terms In all cases, the low energy effective lagrangian for the observable sector develops what is so called: soft supersymmetry breaking terms

2 ∗ 3 Lsusy,soft = mφφ φ + (Mλλλ + h.c.) + (Aφ + h.c). (3.23) i.e. masses and trilinear couplings for the scalars, and mass terms for the gauginos.

58 Bibliograf´ıa

[1] Philip Argyres, An Introduction to Global Supersymmetry, http://www.physics.uc.edu/ argyres/661/index.html

[2] N. Seiberg, Naturalness versus supersymmetric non-renormalization theorems, http://arxiv.org/abs/hep-th/9309335.

[3] Matthew Strassler, An Unorthodox Introduction to Supersymmetric Gauge Theory, http://arxiv.org/abs/hep-th/0309149

[4] K. Konishi and K. Shizuya, Nuovo Cim.90A, 111 (1985)

[5] Pierre Bin´etruy, Supersymmetry, Teory, Experiment, and Cosmology, Oxford Univ. Press. 2006. Bibl. F´ısica: A10-640.

[6] Timothy J. Hollowood, 6 Lectures on QFT, RG and SUSY, http://arxiv.org/abs/0909.0859